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Research Papers

Wave Directionality in Three-Dimensional Periodic Lattices OPEN ACCESS

[+] Author and Article Information
Alireza Bayat

Department of Civil Engineering,
Johns Hopkins University,
3400 N Charles Street,
Latrobe Hall 205,
Baltimore, MD 21218
e-mail: abayat1@jhu.edu

Stavros Gaitanaros

Department of Civil Engineering,
Johns Hopkins University,
3400 N Charles Street,
Latrobe Hall 201,
Baltimore, MD 21218
e-mail: stavrosg@jhu.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 21, 2017; final manuscript received October 23, 2017; published online November 13, 2017. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 85(1), 011004 (Nov 13, 2017) (17 pages) Paper No: JAM-17-1527; doi: 10.1115/1.4038287 History: Received September 21, 2017; Revised October 23, 2017

This work focuses on elastic wave propagation in three-dimensional (3D) low-density lattices and explores their wave directionality and energy flow characteristics. In particular, we examine the dynamic response of Kelvin foam, a simple-and framed-cubic lattice, as well as the octet lattice, spanning this way a range of average nodal connectivities and both stretching-and bending-dominated behavior. Bloch wave analysis on unit periodic cells is employed and frequency diagrams are constructed. Our results show that in the low relative-density regime analyzed here, only the framed-cubic lattice displays a complete bandgap in its frequency diagram. New representations of iso-frequency contours and group-velocity plots are introduced to further analyze dispersive behavior, wave directionality, and the presence of partial bandgaps in each lattice. Significant wave beaming is observed for the simple-cubic and octet lattices in the low frequency regime, while Kelvin foam exhibits a nearly isotropic behavior in low frequencies for the first propagating mode. Results of Bloch wave analysis are verified by explicit numerical simulations on finite size domains under a harmonic perturbation.

There have been a continuously growing number of studies on the dynamic behavior of architected materials with a view toward the design of novel material systems with unique properties for acoustic and vibration mitigation applications [13]. The focus has been mainly on periodic materials and structures known as phononic materials that exhibit bandgap properties; that means the existence of frequency interval(s) within which wave propagation through the material is prohibited [4,5]. These phenomena are mainly correlated to wave interference, so called Bragg scattering or local resonance of microstructural elements.

Two-dimensional (2D) periodic lattice materials have attracted significant interest due to the wide range of properties and functionalities one can achieve by modifying their underlying topology and/or the base constituent material. Several studies have utilized Bloch's theorem and finite element analysis to examine the phononic bandgap properties in triangular, regular, and re-entrant honeycomb, square, and Kagome lattices [68]. Complete bandgaps were shown to exist for a wide range of relative densities for the triangular lattice compared to the hexagonal honeycomb [6]. In Ref. [7], the in-plane elastodynamic behavior of hexagonal lattices was tailored by introducing chirality into the topology of the microstructure. Adding internal resonators in rectangular and hexagonal frames is shown to generate a locally resonant structure providing high frequency bandgap properties around the corresponding natural frequencies [9]. In another study, internal struts are added to the triangular and hexagonal lattices as oscillators providing tunable phononic bandgap properties [10].

Phononic crystals are also shown to have frequency-dependent wave beaming characteristics with applications in wave attenuation and guiding [1113]. Wave directionality of lattices is produced by partial wave filtering in certain spatial directions forcing waves to propagate in the remaining angular directions. Wave directionality phenomena have been extensively investigated in 2D square, triangular and hexagonal lattices for various relative densities and microstructural topologies [6,1113]. While most low-density lattices with straight struts have not shown bandgap properties for a wide range of relative densities [6], introducing curvature in the lattice's beam struts to generate so-called undulated lattices generated complete and partial bandgaps in certain spatial directions [14].

Several techniques have been developed to represent beam steering or directionality characteristics of periodic lattices. Among them are constant-frequency contour plots, phase and group velocity plots, polar group velocity contours, and direct transient simulations on finite size domains [1115]. The phase and group velocities have also been utilized as a measure of the material stiffness in specific directions, revealing anisotropy of a lattice structure at a specific frequency [1,2,16]. An equivalent continuum model is developed in Ref. [17] to explore phase speed and dispersion relations in a three-dimensional (3D) periodic truss structure taking into account inertial effects. A recent work introduced a new topology consisting of hollow tubes interconnected with solid and hollow joints to generate and tailor bandgaps in the structure [18] while wave transmission properties in 3D solid structures are studied in Refs. [19] and [20]. In a recent effort, the combination of local resonance with structural modes was utilized to design a 3D periodic structure that controls and widens bandgaps in low frequencies [21].

Even though wave propagation phenomena have been widely investigated in 2D lattices, to date wave beaming and directionality in 3D low-density lattices have been largely understudied. Here, we focus on the linear wave propagation characteristics of regular 3D lattice materials in order to examine their dispersive directionality behavior. Bloch wave and finite element analysis are employed to derive dispersion surfaces, and estimate phase and group velocities for Kelvin foam, an octet lattice, as well as simple and framed cubic lattices. Our analysis includes new depiction of iso-frequency contours in wavenumber space and novel representation of group velocity plots for 3D structures, providing a frequency-dependent scatter polar plot spanning all spatial directions. Explicit numerical simulations are performed on finite size domains of the considered structures and results are compared with the unit cell analysis.

Figure 1 represents four 3D lattices considered in this work, to study the linear wave propagation phenomena based on their topological and mechanical properties. In a way, they are a direct extension of the 2D lattices examined in Ref. [6] that combine both stretching-and bending-dominated materials. Included in Fig. 1 are the corresponding unit cells of each lattice in canonical form (lower left cells in Fig. 1(a)1(d)) and the one chosen for Bloch wave analysis (unit cells represented in right column in Fig. 1). The Kelvin foam, a body-centered cubic structure, is constructed from the periodic assemblage of the tetrakaidecahedron cell, that is, a fourteen-sided polyhedron consisting of eight hexagonal faces, four rhombic faces, and two square faces (lower left cell shown in Fig. 1(a)). The Kelvin cell was initially thought of as the 3D space-filling structure with the minimal surface area until the Weaire–Phelan foam was shown (but not mathematically proven) to have a smaller area [22,23]. The unit cell chosen for analysis here (Fig. 1(a)) consists of 24 struts on six faces and has been used for the analysis of open-cell foams, including linear elastic properties, bifurcation, and limit loads [2426]. A simple cubic lattice and corresponding unit cells are shown in Fig. 1(b). The octet lattice is a face-centered cubic structure consisting of the periodic placement of an octahedral cell enclosed by eight tetrahedra as shown in Fig. 1(c). Additionally, to investigate the wave propagation phenomena on a highly stretching-dominated lattice, we consider a framed-cubic lattice consisting of two diagonal ligaments added to each faces of the simple cubic lattice. The framed-cubic lattice and corresponding unit cell are shown in Fig. 1(d). Relative density of lattices is defined by total volume of the ligaments divided by the unit cell's volume. Struts have a constant circular cross-sectional area, A, computed from the relative density relation for each lattice. Linear elastic material properties used are: ρ = 2700 kg m−3, E = 71 GPa and ν = 0.33, where ρ, E, and ν are density, Young's modulus, and Poisson's ratio values corresponding to aluminum.

Each unit cell is characterized by three basis vectors, a1, a2, and a3, which lie along the x, y, and z coordinate axes. The Kelvin unit cell consists of six identical square frames forming a cubic lattice with direct lattice vectors a1=(22L,0,0),a2=(0,22L,0), and a3=(0,0,22L) The direct lattice vectors for the octet lattice are a1=(2L,0,0),a2=(0,2L,0), and a3=(0,0,2L) and those of both simple- and framed-cubic lattice are a1=(L,0,0),a2=(0,L,0), and a3=(0,0,L), where L is the strut length. The infinite periodic material is described by a primitive unit cell, U and spatially traversed through lattice vectors a1, a2, and a3 so that the position of any point p within an arbitrary cell of the infinite periodic lattice can be expressed by Display Formula

(1)rp=r0+n1a1+n2a2+n3a3

where rp is the position vector of point pin global coordinates, r0 corresponds to the local position vector of point pin the primitive cell U, and niZ are integers indicating the cell location within the infinite lattice. While a variety of different approaches exist for the study of wave propagation and band structure calculations for phononic materials, the most common ones are the plane wave expansion method, the finite element method, and the finite difference time domain method (see Ref. [4] and references therein). The plane wave expansion method requires a large number of plane waves to scan a 3D space for wave beaming analysis. Finite differences are also computationally expensive and not well suited for complex topologies. Here, we study linear elastic waves in periodic structures using the finite element method and by applying Bloch's theorem [27] to connect the displacement u(rp) of any point p in the lattice to the displacement u(r0) of the corresponding point in primitive unit cell U through Display Formula

(2)u(rp)=u(r0)eik.(rpr0)

where k is the wave vector. Any spatial function Q within the periodic lattice needs to satisfy the periodicity condition Q(r0+R)=Q(rp) therefore one can deduce that eik.R=1, which requires k.R=2mπ (m is an integer). It can be shown that this condition is only achieved when the wave vector is defined in the reciprocal lattice base k=k1b1+k2b2+k3b3 where Display Formula

(3)b1=2πa2×a3a1(a2×a3),b2=2πa3×a1a1(a2×a3),b3=2πa1×a2a1(a×2a3)
so that aibj=2πδij. The propagation of linear elastic waves is studied on the primitive cell domain using finite element method by seeking time-harmonic plane wave solution u(r0)=u0eiωt of the wave equation, where ω is the angular frequency and u0 is the amplitude. All struts are discretized using Timoshenko beam elements (B32) in abaqus. Element matrices are assembled to form the global unit cell mass and stiffness matrices M and K, respectively. Hence, harmonic wave motion of the material at the frequency ω is expressed by Display Formula
(4)(Kω2M)u0=f

where u0 and f are the nodal displacement vector and force vector, respectively. Bloch-type displacement and traction boundary conditions are applied on the opposite boundary nodes of the primitive unit cell. Hence, the displacement and traction forces of nodes periodically located on the faces of a cubic unit cell are related through Display Formula

(5)u=rightc1u,leftu=topc2u,bottomufront=c3uback
Display Formula
(6)fright=c1f,leftftop=c2fbottom,ffront=c3fback

where ci=ei2πki. The complex valued displacement fields resulting from the Bloch condition in Eqs. (5) and (7) cannot be dealt directly within abaqus. For this purpose, all fields are split into real and imaginary parts by creating two identical finite element models, coupled by a MPC subroutine following [28]. We note that for standard finite element solvers, once the displacement boundary conditions (5) are applied, the traction boundary conditions in Eq. (6) are automatically fulfilled. Using Eqs. (5) and (6) in Eq. (4) results in a reduced eigenvalue problem [28] Display Formula

(7)[Kr(k)ω2Mr(k)]ur=0

where Kr and Mr are stiffness and mass matrices corresponding to both real and imaginary meshes and the reduced displacement field vector, ur. For any given k vector, the eigenvalues ω=ω(k) will be calculated by solving (7). Due to the periodicity of the reciprocal lattice, the dispersion surfaces can be obtained by confining the k vectors to the first Brillouin zone (FBZ). Dispersion surfaces, also referred to as phase constant surfaces, demonstrate the frequency–wavenumber relation for a single mode of propagating wave and can be used to investigate the frequency of free wave motion of the periodic structure for any k vector. While the computations of 3D phase constant surfaces require evaluating the eigenvalues of Eq. (7) for all combinations of triplets (k1, k2, k3) (wave vector components in x, y, and z directions), covering the FBZ domain, dispersion relations can also be plotted as band diagrams. Due to symmetry conditions of a cubic direct lattice, the analysis can be further limited to the irreducible Brillouin zone (IBZ) as shown in Fig. 1(e). The eigenvalue problem (7) is solved by sweeping the k vector on the perimeter of IBZ following G–X–M–G–R–M–X–R, where the coordinates of the IBZ corners are shown in Fig. 1(e). Limiting the wavenumber along the perimeter of the IBZ significantly reduces the computational cost of the frequency response diagram. Bandgaps are evaluated as the gaps between the subsequent branches of the frequency bands and feature the range of the frequencies in which the wave propagation is blocked.

Band diagrams provide a compact representation of consecutive dispersion curves and the associated frequencies, and they can be used to find the complete/partial bandgaps of a periodic structure. Dispersion relations are also represented in the form of iso-frequency contours to fully demonstrate the wave propagation characteristics. For a 3D analysis, the iso-frequency contours will be a 2D representation of frequency of the wave in the (k1, k2), (k2, k3), and (k3, k1) planes for constant values of k3, k1, and k2, respectively. The symmetry of dispersion surfaces allows the iso-frequency plots to be restricted in the FBZ domain. Favored directions of wave propagation, as well as phase and group velocities, can be derived from the same iso-frequency contours. Phase velocity is defined as the speed at which any fixed-phase point of the wave is displaced in terms of frequency and is aligned with the wave vector, i.e., V=(ω/k2)k, where k = |k|. It is calculated from constant frequency plots by finding the set of k vectors that correspond to a prescribed frequency ω. The set of wave vectors is then used to plot the angular variation of phase velocity in terms of wave vector components. In a nondispersive medium, the phase velocity is independent of wavenumber and the phase velocity plot yields a circle with a constant radius independent of frequency. Group velocity represents the speed of a wave envelope and lies in the direction normal to the corresponding iso-frequency contour so that c=ω/k. For a given frequency, the directions of energy flow within the structure are identified by the normal to the corresponding iso-frequency contour lines. Hence, preferential or prohibited directions of wave propagation can be explored in terms of group velocity plots. The dispersive behavior of a material can also be realized by comparing the group and phase velocity plots since for a nondispersive wave, the group velocity is the same as the phase velocity. Phase and group velocities can also be used as a compact representation of the effective anisotropic nature of the structure for a given frequency. For a solid isotropic medium, the phase and group velocities show a circular curve in which the wave propagates in same speed in all directions. Deviation of the velocity curves from a circle is used as a measure of anisotropy of the structure for a given frequency.

Kelvin Foam.

Dynamic analysis of the lattices described in Sec. 2 is performed using the finite element method on a unit cell domain. The dynamic response of the Kelvin unit cell with relative density ρr = 15% is shown in Fig. 2. Band diagrams are plotted as normalized frequency Ω=ωa/ω0 versus wavenumber k, where a=(|a1|+|a2|+|a3|)/3 and ω0=π2EI/ρL4 are the average characteristic lattice size and the first flexural resonance frequency of a pinned–pinned beam respectively, and I is the second moment of the strut's cross-sectional area. Figure 2(a) shows the band diagram of the first ten dispersion curves resulting from solving the eigenvalue problem (7). No complete bandgaps appear for this lattice structure within the first ten modes. A partial bandgap exists at Ω = 1.21 in the X–M direction. Figure 2(b) shows the density of states, normalized by its maximum value, corresponding to the first ten dispersion surfaces of the structure. The density of states (also known as modal density) at a given frequency, D(ω), is defined as the number of states (k vectors) per unit frequency range ω + , given by Kittel [27] Display Formula

(8)D(ω)=(L2π)3dSω|c|

where the integration is taken over all surfaces Sω at a given frequency ω in the wavenumber space. Density of states are numerically computed by solving the eigenvalue problem (7) for 9261 (213) combinations of (k1, k2, k3) states over the FBZ and summing up all existing states in the specific range of frequency. The eigenstates are collected for all values of the k vector into bins of width ΔΩ, where ΔΩ depends on the numerical sampling of the (k1, k2, k3) space. The plot is shown as a histogram in Fig. 2(b). While fewer modes are excited in the low frequency regime, higher magnitude of modal density is occurring in the frequency range 0.7 < Ω < 1.6. Local maxima of the density of states plot occur at the points where minimum or maximum bounding frequencies of each dispersion surface since the modal density is directly proportional to dk/dω; thus, at the points where the slope of the dispersion surface is zero, the D(ω) tends to infinity. Here, due to the numerical implementation of the density of states, the state count remains finite. The peak of the modal density happens at the points where several consecutive dispersion surfaces are tangent to each other, Ω = 1.32. As it is expected from the band diagram plot, no bandgap appears in the density of states plot. The first three eigenmodes of the Kelvin cell at the symmetry points X, M, and R of the FBZ are shown in Fig. 2(c). At X and M, the first two modes show local out-of-plane bending of struts, while the third mode indicates a stretching behavior in the [−1 1 0] direction. At symmetry point R, the same localized out-of-plane bending of ligaments is happening while the unit cell deformation is dominated by a tilting of the frames in the x–z plane.

For 3D lattice structures, the iso-frequency contour plots are more conveniently represented in different planes of the wavenumber space. Figure 3 shows the dispersion surfaces of the Kelvin cell as 2D frequency-constant contours in the positive quadrant of the ky − kz plane for different magnitudes of the kx > 0 component. Due to symmetries of the Kelvin cell, only contours for ky − kz planes are presented. The contour plots shown for the first three modes correspond to the first three bands in Fig. 2(a). For classical bulk solids, the first and second modes are designated by shear or transverse modes and correspond to two different polarizations while the third mode is referred as longitudinal or compressive mode. However, the shear and longitudinal terminology is not completely meaningful for the case of 3D lattice structures considered here, since the mode shapes represent a combination of shear, stretching and bending in the ligaments for all propagating modes. For a given frequency, a harmonic point load in a lattice will excite those propagating waves (k1, k2, k3) that lie on the corresponding contour and the energy flow will be in the direction of the normal to the contour curve. Circular contours suggest an isotropic behavior for the lattice in the corresponding frequency range. The group velocity vector is normal to the frequency-constant contour and in the direction of increasing frequency. For kx = 0 and at low frequencies—where the contours are nearly circular—a quasi-isotropic behavior is expected for the first three wave modes, while by increasing kx, the lattice shows a complex anisotropic behavior at different range of frequencies.

Phase and group velocity diagrams are then extracted from the constant-frequency contours to examine wave directionality and beaming effects. Figure 4 shows the phase and group velocities normalized by their corresponding maximum magnitude for the Kelvin structure at different frequencies. For the sake of brevity, only the plot corresponding to kx = 0 is presented. At Ω = 0.1 and Ω = 0.3, the discrepancy between phase and group velocities indicates an anisotropic and highly dispersive dynamic behavior. At low frequency, Ω = 0.1 the first wave mode propagates with a higher phase velocity along the 0 deg direction while the direction of energy flow mainly lies along ±45 deg for the second and third modes. At the higher frequency Ω = 0.3, increased wave speed is observed for the third mode along 0 deg and 90 deg directions. All modes highlight a pronounced preferential direction of wave propagation along 0 deg. The lobes inside the phase velocity plots suggest that for the short and long wavelengths, the wave propagates at different speeds. In other words, the phase velocity plots highlight the fact that at higher normalized frequencies, Ω = 0.3 and Ω = 0.7, the structure shows anisotropic behavior in the long wavelength limit while a nearly isotropic characteristic in the short wavelength regime. At Ω = 0.7, the plots are close to being circular indicating a quasi-isotropic behavior, which suggests an identical speed of wave propagation in all directions for the first three propagating modes. The contour plots shown in Fig. 3 demonstrate that the structure shows more anisotropic characteristic by further increasing values of kx.

We introduce a scatter plot in Fig. 5 to demonstrate a compact four-dimensional (4D) representation of group velocity versus frequency and angular directions in spherical coordinates. The radial coordinate corresponds to the normalized frequency Ω, the direction of group velocity is pictured in full angular directions θ and φ, and the color code is assigned to each individual scatter point to denote the magnitude of the group velocity normalized by its maximum value. The resolution of the plots depends on the number of divisions of the reciprocal lattice axes in the FBZ domain. Results shown in Fig. 5 are computed for 21 × 21 × 21 states in the wavenumber space. Clearly, increasing the number of states further improves the accuracy of the plot but with a higher computational cost. Preferential wave propagation directions of the first mode are observed in Ω < 1 and 60 deg < θ < 90 deg. No significant wave directionality is observed for second and third modes. Also, shown in Fig. 5 is the existence of directional bandgaps for all modes in the frequency range1 < Ω < 1.5. Although previous studies on two-dimensional beam lattices suggest that complete bandgaps arise within the first resonance frequency of a pinned–pinned beam, Fig. 5 indicates that partial bandgaps are not consistent with the resonance frequency of beam as confirmed in Ref. [12].

Simple-Cubic Lattice.

The dynamic response of simple cubic lattice with ρr = 15% is shown in Fig. 6. Band diagram shows no complete bandgap but directional bandgaps exists at Ω = 0.8 and at Ω = 1.2 in X–M and R–M directions, respectively. Low density of states (Fig. 6(b)) appears in the low frequency regime while high modal densities happen in the range 1.2 < Ω < 1.4. Figure 6(c) represents the first three eigenmodes of the simple cubic cell at high symmetry points of FBZ. We note that the cells shown here are in canonical form and are different than the unit cell used for Bloch wave analysis. First mode shows local stretching of ligaments for symmetry points, X, M, and R, while the second and third modes indicate local bending of struts for the high symmetry points M and R.

Iso-frequency contours of the cubic lattice in ky − kz plane (ky, kz > 0) for increasing kx are shown in Fig. 7. In kx = 0 plane, all modes exhibit isotropic behavior in low frequencies while the lattice behaves anisotropic for the first three modes in higher frequencies. For higher values of kx, the iso-frequency contour patterns predict distinct directional characteristic for all three wave modes, particularly along ky and kz axes directions. Phase and group velocity plots of the simple cubic lattice corresponding to the ky − kz contours at kx = 0 plane are shown in Fig. 8 for three selected frequencies covering the range of frequencies observed in dispersion contours. The vast differences between the phase and group velocity curves imply a highly dispersive behavior of the cubic lattice for all three modes of propagating waves. At Ω = 0.2, the structure represents a directionality along horizontal direction for the first propagating mode, while a nearly isotropic pattern for the second and third propagating modes. At higher frequencies, Ω = 0.4 and Ω = 0.6, the structure exhibits a strong directional behavior in horizontal direction, for the first and second modes. In contrast, the third mode conserves a nearly isotropic behavior a Ω = 0.4 and turns it to a directional pattern along vertical direction at higher frequency Ω = 0.6.

Figure 9 shows 4D polar velocity plots for the simple cubic lattice. A significant partial bandgap is observed in 0.2 < Ω < 0.8 in orientations 10 deg < θ < 80 deg and 10 deg < φ < 80 deg for the first two wave modes. Strong wave beaming is observed along θ ≈ 0 deg, θ ≈ 90 deg, and φ ≈ 90 deg directions for the first three modes. One may notice that in 0.2 < Ω < 1.0 range, the magnitude of the group velocity is negligible in the 10 deg < θ < 80 deg and 10 deg < φ < 80 deg directions, suggesting preferential directions for wave propagation for all three modes.

Octet Lattice.

The dynamic response of an octet lattice with ρr = 15% is shown in Fig. 10. No bandgap appears in the band diagram within the first ten modes. Figure 10(b) shows the density of states corresponding to the first ten dispersion surfaces of the structure. Minor modal density is reported in the low frequency and long wavelength regime, that is below Ω ≈ 8, while the peak of the modal density is at Ω ≈ 11. The lowest three eigenmodes of the octet cell at high symmetry points X, M, and R of the FBZ are given in Fig. 10(c). At X, deformation of the first two modes shows local stretching in central octahedral cell struts and a local buckling of struts in tetrahedrons, while the third mode deformation indicates both local stretching and bending of struts in central octahedron and tetrahedrons. At symmetry point M, the localized bending of ligaments is dominated in both octahedron and tetrahedrons for the first three propagating modes. At symmetry point R, deformation of first and second eigenmodes includes a local buckling and tilting in tetrahedrons, while the central octahedron shows minor deformation. The third mode shows local bending and stretching in all central and boundary struts.

Iso-frequency contour plots of the first three modes for octet lattice in ky − kz plane versus increasing values of kx are given in Fig. 11. Iso-frequency contours represent more sophisticated patterns compared to the Kelvin and simple cubic lattices. By increasing kx and the frequency, more complex contour curves are observed, indicative of the anisotropic and highly dispersive behavior of the lattice. Phase and group velocities of the octet structure corresponding to kx = 0 contours at different frequencies are shown in Fig. 12. Looking at the phase velocity plots at Ω = 4, Ω = 6, and Ω = 8, noticeable anisotropy is observed in short wavelength vicinity in vertical direction, while we observe anisotropy in horizontal direction for long wavelength limit. Group velocity plots exhibit strong directionality in vertical directions for the first two modes of the three considered frequencies. All three modes represent same directional behavior for intermediate frequencies Ω = 6.

Group velocity scatter plots for the octet lattice are shown in Fig. 13 for the first three modes. As it is expected from the density of states plot, gross distribution of the scatterers happens above Ω ≈ 10. We note that the magnitude of the group velocity is insignificant around Ω ≈ 10 due to the nearly flat dispersion curves (Fig. 10(a)) in this range of frequency. On the other hand, distribution of scatter points is somewhat sparse below this frequency, henceforth providing partial bandgaps in different spatial directions of θ and φ, and for the first three propagating modes.

Framed-Cubic Lattice.

Previous works have studied the link between the average connectivity of 2D beam lattices to their bandgap behavior [29] considering the locally resonant phenomenon. Here, the Kelvin, simple-cubic, and octet lattices characterized by average connectivity of 4, 6, and 12 do not show any bandgap property in low frequencies. To further explore the effect of connectivity in 3D lattices, we investigate the dynamic response of a so-called framed-cubic lattice which is featured with a high connectivity number, 18. High connectivity provides a strong stretching-dominated behavior for the lattice in the static response. Figure 14 reports the band diagram and density of states of the framed-cubic lattice. A bandgap appears at Ω ≈ 2.25. Large phase speed in long wavelength region happens due to the high stiffness of the structure. The maximum of density of states is located just below Ω ≈ 2.25, due to flat regions of the dispersion surfaces. Iso-frequency contours and group velocity plots are not shown here for the sake of brevity.

Effect of Topological Anisotropy.

We next consider the effect of an imposed geometric anisotropy of the lattices on their wave propagation characteristics by increasing the size of the characteristic unit cell in one direction by a factor λ = 1.2. Therefore, struts with a projection in the y-direction are elongated and the lattice vector a2 is also enlarged by 20%. The unit cell and the corresponding reciprocal lattice for the anisotropic cell will both now be tetragonal (instead of cubic). Results show that the geometric anisotropy of lattices has a minor effect on the band diagrams while the directional behavior is, as expected, influenced for the three considered lattices. Here, we only present the result of the anisotropic Kelvin lattice. Figure 15 shows frequency-constant contours for the anisotropic Kelvin cell in the positive quadrant of the ky − kz plane for different magnitudes of kx. Comparison of contour patterns with the isotropic Kelvin cell demonstrates that for low frequencies, both lattices possess isotropic behavior for the first three propagating modes. For higher frequencies, the direction of energy flow is conspicuous along the ky direction for the anisotropic cell. On the other hand, contours in the kx − kz plane (not shown here) are not noticeably altered, due to the preserved symmetry of the lattice in the x − z plane. Figure 16 shows the effect of anisotropy on the group velocity of the Kelvin lattice. The denser population of scatterers and the higher magnitude of group velocity along φ ≈ 90 reveal that the group velocity is enhanced in the elongation direction for the first three propagating modes, while the energy flow is nonsignificant in 0 deg < φ < 30 deg direction. A partial bandgap appears in 0 < Ω < 0.5 and 60 deg < θ < 90 deg and 0 deg < φ < 15 deg for the third mode of propagation.

Dynamic Anisotropy Index.

Phase and group velocity diagrams have been extensively utilized to demonstrate the degree of anisotropy of the lattices in dynamic regimes. For 2D lattices, an anisotropy index has been introduced in Ref. [16] using the standard deviation of the phase velocity. Dynamic anisotropy has also been measured by the standard deviation of group velocity diagrams, which exhibits the discrepancy in velocity of wave envelopes in different directions [14]. When the group velocity plot is circular, no preferential direction of energy flow is observed, revealing a zero anisotropy index of the lattice while a nonzero anisotropy index represents a degree of dynamic anisotropy. Here, we extend the anisotropy index for 3D lattices by integrating the standard deviation of the phase velocity in the ky − kz plane over different magnitudes of the kx component of the wave vector spanning the full 3D wavenumber space Display Formula

(9a)AI(Ω)=02π|a1|σ(V(Ω))dk

where Display Formula

(9b)σ(V(Ω))=02π[V(θ,Ω)V¯(Ω)V¯(Ω)]2dθ
is the standard deviation of the phase velocity magnitude at a specific frequency and V¯ is the average value of the phase velocities' magnitudes. We note that the anisotropy index is a measure of directionality in wave propagation and highlights any anisotropy in the dynamic behavior of the lattice. Figure 17 reports the anisotropy index versus normalized frequency for the Kelvin, simple cubic and octet lattices. The figure demonstrates the anisotropic index of the considered lattices for relative density ρr = 2%, ρr = 15% and topological asymmetry (elongation in y direction). The response of lattices with low relative density covers a wider range of frequency spectrum for the first three modes. For the Kelvin lattice, the anisotropy index has an ascending trend for increasing frequency for both low and high relative densities. The structure exhibits high anisotropy at elevated frequencies. The three considered lattices show less dynamic anisotropy for ρr = 15% for the first three propagating modes.

For Kelvin and simple cubic lattices, the third mode shows less dynamic anisotropy compared to the first two propagating modes. The octet lattice indicates a different behavior compared to the Kelvin and simple cubic lattices. The low density, ρr = 2% octet lattice, demonstrates a descending trend versus increasing frequency. The lattice possesses high dynamic anisotropy at low frequencies, whereas it tends to a more isotropic characteristic at high frequencies. For the octet lattice with ρr = 15%, the dynamic anisotropy decreases by increasing frequency for <4, while it displays an ascending trend for higher frequencies. Despite the Kelvin and cubic lattices, the third mode exhibits higher anisotropy for symmetric and asymmetric octet lattices with ρr = 15%. Topological asymmetry exhibits a pronounced effect on the dynamic anisotropy of the lattices. Lattices show high dynamic anisotropy specifically at elevated frequencies.

Although Bloch analysis is valid for the infinite periodic structure, it is always useful to compare results concerning directionality with direct transient simulations on finite size domains as suggested by Trainiti et al. [14], Ruzzene et al. [15], and Casadei and Rimoli [16]. The finite size lattice is excited by a point harmonic perturbation load at a center node. Such perturbation excites the lattice at a prescribed frequency and generates wave envelopes carrying the energy of the wave through the lattice. The transient response of the lattice has been evaluated by direct numerical integration of the equations of motion [14,15]. The wave envelope patterns are presumably in the direction of the maxima in the group velocity plots. Response of a periodic lattice to a point harmonic loading has also been verified by a modal expansion technique [7]. Attention is focused on how to feature each mode of propagation individually, since when multiple modes are present at an excitation frequency, the displacement contours will be intricate due to the multiple directionality of propagating modes. The well-known Helmholtz decomposition is utilized to distinguish the transverse and longitudinal modes in 2D lattices. In this approach, curl and divergence of the displacement contours are taken to visualize longitudinal and transverse propagating modes, respectively, which requires further post-processing of the direct simulation results [16]. In another attempt, to avoid complex response of the structure due to interference of multiple modes, the structure is excited at a frequency that only a single mode is present (according to the band diagram) and displacement contours verified the group velocity of corresponding mode [14]. This technique can also be employed at certain excitation frequencies where multiple modes are present but one has zero group velocities corresponding to the inflection points of the dispersion surfaces [14]. Things are more complicated for 3D structures due to the presence of three wave modes in the low-frequency regime.

In the lattices studied herein, since multiple propagating wave modes are present in the low frequency region, we seek the excitation frequencies in which they have similar group velocity patterns so that the wave envelope visualizes same directionality over the finite size lattice. Figure 18 demonstrates the time-dependent response of a 10 × 10 × 10 Kelvin lattice excited by a point displacement harmonic perturbation. The size of the sample is limited by the computational cost and chosen so that the model has enough cells in order to allow visualizing the wave envelopes before waves reach the boundary faces. Explicit dynamics analysis is adopted for numerical simulations, which performs central-difference time integration on the wave equation through a large number of infinitesimal time increments. The central-difference procedure solves the wave equation at the beginning of each time increment t, utilizing a lumped element mass matrix and is fairly cost-effective compared to the direct-integration operation. The center node of the lattice in y − z plane (x = 0) is perturbed by a harmonic displacement point load at a normalized frequency Ω = 0.7. The amplitude of excitation is chosen so that a linear elastic wave motion is maintained in the analysis. The iso-frequency and group velocity reveal a nearly circular pattern at frequency Ω = 0.7 as was expected based on the results shown in Figs. 35. All nodes on the boundary faces parallel to y − z plane are assigned displacement-free boundary conditions while the translational and rotational degrees-of-freedom of all other outer boundary nodes are held fixed.

In order to capture the propagating wave patterns, careful consideration should be paid to the selection of explicit central-difference time-step and largest element size in the ligaments correspond to the excitation frequency. A general rule of thumb is to select the largest element size between 10 and 15 times less than smallest wavelength. Figure 18 demonstrates front and isometric views of total displacement contours resulting from transient analysis. The contour patterns verify a nearly circular propagating wavefront in the finite domain Kelvin lattice by increasing step times. No preferential directions of wave propagation are observed reinforcing the results reported in Figs. 4 and 5.

Transient analysis simulation of a simple cubic lattice is similarly performed through explicit dynamics analysis on a 20 × 20 × 20 finite domain. The center node of the model in yz plane (x = 0) is perturbed by a harmonic displacement point load. To avoid complex wave envelope patterns, we seek certain excitation frequencies in which the group velocity plots show similar directionality for all three propagating modes. Group velocity plots in Fig. 9 show marked directionality at normalized frequency Ω = 0.4 and in θ = 0 deg, θ = 90 deg, and φ = 90 deg directions for first, second, and third mode. Interestingly, these directions lie along the ligaments in a simple cubic lattice. Total displacement contour snapshots of transient response at Ω = 0.4 are shown in Fig. 19. The finite size structure reveals a distinct directionality along lattice ligaments as is predicted by group velocity plots.

A 10 × 10 × 10, finite size model of octet lattice is considered for transient analysis. The lattice is perturbed by a harmonic point displacement at normalized frequency Ω = 7 and on center node in y − z plane (x = 0). In Fig. 13, although scatterers are thinly distributed in nearly multiple spatial directions but narrow wave beaming is observed in 1 < Ω < 9, specifically in θ = 0, θ = 45, and θ = 90 directions and for the first three propagating modes. We note that due to symmetry of the lattice, the wave beaming also happens in φ = 0, φ = 45, and φ = 90 spatial direction. Figure 20 demonstrates the transient response in octet lattice by increasing step times. As suggested by group velocity plots, slight wave beaming is detected in θ = 0, θ = 45, and θ = 90 directions.

Lattice materials have been known for their unique properties that are derived from the intricate interplay between topology and base material behavior. Here, we present a complete analysis of the elastic wave propagation characteristics of 3D low-density periodic lattice materials with different topological characteristics and identify key features of their dynamic behavior. In particular, we focus on the Kelvin foam, the octet lattice, and the simple- and framed-cubic lattices. Frequency band diagrams are constructed using Bloch analysis on representative unit cells. Our results indicate that for low relative densities analyzed here (2% < ρr < 15%), the framed-cubic lattice shows a complete bandgap property, which also is featured with the largest connectivity number. In contrast, the band diagrams for the Kelvin, simple cubic, and the octet lattice did not show any full bandgaps, which is interesting noting that previous studies on 2D lattices reported bandgap behavior in the dynamic response of both honeycomb and triangular lattices. Evidently, findings on 2D beam lattices cannot be extended to provide any insight on the responses of 3D counterparts due to the more complex wave interference in three dimensions.

New representation of iso-frequency contours and group velocity plots is introduced to demonstrate directionality and wave propagation characteristics for each structure. Iso-frequency contours are developed in the 3D wavenumber space to examine the dispersive isotropy or anisotropy of elastic waves propagating in each lattice material. Our results show that for long wavelength waves, Kelvin foam exhibits a nearly isotropic behavior in low frequencies and for the first propagating mode, while in high frequencies, complex contour patterns accompanied by narrow partial bandgaps are found for the second and third modes. Wave beaming is investigated through frequency-dependent group velocity spherical plots in 3D space providing a useful tool for seeking preferential directions of energy flow in 3D lattices. The simple cubic lattice revealed pronounced wave beaming along the unit cell edges' directions while narrow wave beaming is observed in multiple directions for the octet lattice. These results from unit cell analysis are verified by transient simulation on finite-size lattices.

The effects of relative density and topological asymmetry on the wave propagation are also explored. Topological anisotropy shows to influence the group velocity plots and partial bandgaps of the Kelvin lattice. Finally, a frequency-dependent anisotropy index is defined for 3D lattices to quantify the degree of wave directionality of structure. For the Kelvin and simple cubic lattices, the anisotropy index has an ascending trend by increasing frequency for both low and high relative densities. The Kelvin, simple cubic, and octet structures with ρr = 15% exhibit high anisotropy at elevated frequencies. The three considered lattices show less dynamic anisotropy for higher relative density and for the first three propagating modes. The octet lattice, featured as a stretching-dominated lattice, represents more complex dynamic anisotropy index by increasing frequencies.

This research was supported by Johns Hopkins University and the Maryland Advanced Research Computing Center (MARCC). Their support is acknowledged with thanks.

Mead, D. M. , 1996, “ Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964–1995,” J. Sound Vib., 190(3), pp. 495–524. [CrossRef]
Langley, R. S. , 1994, “ On the Modal Density and Energy Flow Characteristics of Periodic Structures,” J. Sound Vib., 172(4), pp. 491–511. [CrossRef]
Brillouin, L. , 1953, Wave Propagation in Periodic Structures, 2nd ed., Dover Publications, Mineola, NY.
Hussein, M. I. , Leamy, M. J. , and Ruzzene, M. , 2014, “ Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook,” ASME Appl. Mech. Rev., 66(4), p. 040802. [CrossRef]
Sigalas, M. , and Economou, E. N. , 1993, “ Band Structure of Elastic Waves in Two Dimensional Systems,” Solid State Commun., 86(3), pp. 141–143. [CrossRef]
Phani, A. S. , Woodhouse, J. , and Fleck, N. A. , 2006, “ Wave Propagation in Two-Dimensional Periodic Lattices,” J. Acoust. Soc. Am., 119(4), pp. 1995–2005. [CrossRef] [PubMed]
Spadoni, A. , Ruzzene, M. , Gonella, S. , and Scarpa, F. , 2009, “ Phononic Properties of Hexagonal Chiral Lattices,” Wave Motion, 46(7), pp. 435–450. [CrossRef]
Gonella, S. , and Ruzzene, M. , 2008, “ Analysis of in-Plane Wave Propagation in Hexagonal and Re-Entrant Lattices,” J. Sound Vib., 312(1), pp. 125–139. [CrossRef]
Baravelli, E. , and Ruzzene, M. , 2013, “ Internally Resonating Lattices for Bandgap Generation and Low-Frequency Vibration Control,” J. Sound Vib., 332(25), pp. 6562–6579. [CrossRef]
Martinsson, P. G. , and Movchan, A. B. , 2003, “ Vibrations of Lattice Structures and Phononic Band Gaps,” Q. J. Mech. Appl. Math., 56(1), pp. 45–64. [CrossRef]
Langley, R. S. , Bardell, N. S. , and Ruivo, H. M. , 1997, “ The Response of Two-Dimensional Periodic Structures to Harmonic Point Loading: A Theoretical and Experimental Study of a Beam Grillage,” J. Sound Vib., 207(4), pp. 521–535. [CrossRef]
Zelhofer, A. J. , and Kochmann, D. M. , 2017, “ On Acoustic Wave Beaming in Two-Dimensional Structural Lattices,” Int. J. Solids Struct., 115–116, pp. 248–269. [CrossRef]
Wang, Y. F. , Wang, Y. S. , and Zhang, C. , 2014, “ Bandgaps and Directional Properties of Two-Dimensional Square Beam-like Zigzag Lattices,” AIP Adv., 4(12), p. 124403. [CrossRef]
Trainiti, G. , Rimoli, J. J. , and Ruzzene, M. , 2016, “ Wave Propagation in Undulated Structural Lattices,” Int. J. Solids Struct., 97–98, pp. 431–444. [CrossRef]
Ruzzene, M. , Scarpa, F. , and Soranna, F. , 2003, “ Wave Beaming Effects in Two-Dimensional Cellular Structures,” Smart Mater. Struct., 12(3), p. 363. [CrossRef]
Casadei, F. , and Rimoli, J. J. , 2013, “ Anisotropy-Induced Broadband Stress Wave Steering in Periodic Lattices,” Int. J. Solids Struct., 50(9), pp. 1402–1414. [CrossRef]
Messner, M. C. , Barham, M. I. , Kumar, M. , and Barton, N. R. , 2015, “ Wave Propagation in Equivalent Continuums Representing Truss Lattice Materials,” Int. J. Solids Struct., 73–74, pp. 55–66. [CrossRef]
Delpero, T. , Schoenwald, S. , Zemp, A. , and Bergamini, A. , 2016, “ Structural Engineering of Three-Dimensional Phononic Crystals,” J. Sound Vib., 363, pp. 156–165. [CrossRef]
D'Alessandro, L. , Belloni, E. , Ardito, R. , Corigliano, A. , and Braghin, F. , 2016, “ Modeling and Experimental Verification of an Ultra-Wide Bandgap in 3D Phononic Crystal,” Appl. Phys. Lett., 109(22), p. 221907. [CrossRef]
Lucklum, F. , and Vellekoop, M. J. , 2016, “ Realization of Complex 3-D Phononic Crystals With Wide Complete Acoustic Band Gaps,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 63(5), pp. 796–797. [CrossRef]
Matlack, K. H. , Bauhofer, A. , Krödel, S. , Palermo, A. , and Daraio, C. , 2016, “ Composite 3D-Printed Metastructures for Low-Frequency and Broadband Vibration Absorption,” Proc. Natl. Acad. Sci. USA, 113(30), pp. 8386–8390.
Thomson, W. , 1887, “ On the Division of Space With Minimum Partitional Area,” London, Edinburgh, Dublin Philos. Mag. J. Sci., 24(151), pp. 503–514. [CrossRef]
Weaire, D. , and Phelan, R. , 1994, “ A Counter-Example to Kelvin's Conjecture on Minimal Surfaces,” Philos. Mag. Lett., 69(2), pp. 107–110. [CrossRef]
Gong, L. , Kyriakides, S. , and Jang, W. Y. , 2005, “ Compressive Response of Open-Cell Foams. Part I: Morphology and Elastic Properties,” Int. J. Solids Struct., 42(5), pp. 1355–1379. [CrossRef]
Gong, L. , and Kyriakides, S. , 2005, “ Compressive Response of Open Cell Foams—Part II: Initiation and Evolution of Crushing,” Int. J. Solids Struct., 42(5), pp. 1381–1399. [CrossRef]
Jang, W. Y. , and Kyriakides, S. , 2009, “ On the Crushing of Aluminum Open-Cell Foams: Part II Analysis,” Int. J. Solids Struct., 46(3), pp. 635–650. [CrossRef]
Kittel, C. , 2005, Introduction to Solid State Physics, 8th ed., Wiley, Hoboken, NJ.
Åberg, M. , and Gudmundson, P. , 1997, “ The Usage of Standard Finite Element Codes for Computation of Dispersion Relations in Materials With Periodic Microstructure,” J. Acoust. Soc. Am., 102(4), pp. 2007–2013. [CrossRef]
Wang, P. , Casadei, F. , Kang, S. H. , and Bertoldi, K. , 2015, “ Locally Resonant Band Gaps in Periodic Beam Lattices by Tuning Connectivity,” Phys. Rev. B, 91(2), p. 020103(R). [CrossRef]
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References

Mead, D. M. , 1996, “ Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964–1995,” J. Sound Vib., 190(3), pp. 495–524. [CrossRef]
Langley, R. S. , 1994, “ On the Modal Density and Energy Flow Characteristics of Periodic Structures,” J. Sound Vib., 172(4), pp. 491–511. [CrossRef]
Brillouin, L. , 1953, Wave Propagation in Periodic Structures, 2nd ed., Dover Publications, Mineola, NY.
Hussein, M. I. , Leamy, M. J. , and Ruzzene, M. , 2014, “ Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook,” ASME Appl. Mech. Rev., 66(4), p. 040802. [CrossRef]
Sigalas, M. , and Economou, E. N. , 1993, “ Band Structure of Elastic Waves in Two Dimensional Systems,” Solid State Commun., 86(3), pp. 141–143. [CrossRef]
Phani, A. S. , Woodhouse, J. , and Fleck, N. A. , 2006, “ Wave Propagation in Two-Dimensional Periodic Lattices,” J. Acoust. Soc. Am., 119(4), pp. 1995–2005. [CrossRef] [PubMed]
Spadoni, A. , Ruzzene, M. , Gonella, S. , and Scarpa, F. , 2009, “ Phononic Properties of Hexagonal Chiral Lattices,” Wave Motion, 46(7), pp. 435–450. [CrossRef]
Gonella, S. , and Ruzzene, M. , 2008, “ Analysis of in-Plane Wave Propagation in Hexagonal and Re-Entrant Lattices,” J. Sound Vib., 312(1), pp. 125–139. [CrossRef]
Baravelli, E. , and Ruzzene, M. , 2013, “ Internally Resonating Lattices for Bandgap Generation and Low-Frequency Vibration Control,” J. Sound Vib., 332(25), pp. 6562–6579. [CrossRef]
Martinsson, P. G. , and Movchan, A. B. , 2003, “ Vibrations of Lattice Structures and Phononic Band Gaps,” Q. J. Mech. Appl. Math., 56(1), pp. 45–64. [CrossRef]
Langley, R. S. , Bardell, N. S. , and Ruivo, H. M. , 1997, “ The Response of Two-Dimensional Periodic Structures to Harmonic Point Loading: A Theoretical and Experimental Study of a Beam Grillage,” J. Sound Vib., 207(4), pp. 521–535. [CrossRef]
Zelhofer, A. J. , and Kochmann, D. M. , 2017, “ On Acoustic Wave Beaming in Two-Dimensional Structural Lattices,” Int. J. Solids Struct., 115–116, pp. 248–269. [CrossRef]
Wang, Y. F. , Wang, Y. S. , and Zhang, C. , 2014, “ Bandgaps and Directional Properties of Two-Dimensional Square Beam-like Zigzag Lattices,” AIP Adv., 4(12), p. 124403. [CrossRef]
Trainiti, G. , Rimoli, J. J. , and Ruzzene, M. , 2016, “ Wave Propagation in Undulated Structural Lattices,” Int. J. Solids Struct., 97–98, pp. 431–444. [CrossRef]
Ruzzene, M. , Scarpa, F. , and Soranna, F. , 2003, “ Wave Beaming Effects in Two-Dimensional Cellular Structures,” Smart Mater. Struct., 12(3), p. 363. [CrossRef]
Casadei, F. , and Rimoli, J. J. , 2013, “ Anisotropy-Induced Broadband Stress Wave Steering in Periodic Lattices,” Int. J. Solids Struct., 50(9), pp. 1402–1414. [CrossRef]
Messner, M. C. , Barham, M. I. , Kumar, M. , and Barton, N. R. , 2015, “ Wave Propagation in Equivalent Continuums Representing Truss Lattice Materials,” Int. J. Solids Struct., 73–74, pp. 55–66. [CrossRef]
Delpero, T. , Schoenwald, S. , Zemp, A. , and Bergamini, A. , 2016, “ Structural Engineering of Three-Dimensional Phononic Crystals,” J. Sound Vib., 363, pp. 156–165. [CrossRef]
D'Alessandro, L. , Belloni, E. , Ardito, R. , Corigliano, A. , and Braghin, F. , 2016, “ Modeling and Experimental Verification of an Ultra-Wide Bandgap in 3D Phononic Crystal,” Appl. Phys. Lett., 109(22), p. 221907. [CrossRef]
Lucklum, F. , and Vellekoop, M. J. , 2016, “ Realization of Complex 3-D Phononic Crystals With Wide Complete Acoustic Band Gaps,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 63(5), pp. 796–797. [CrossRef]
Matlack, K. H. , Bauhofer, A. , Krödel, S. , Palermo, A. , and Daraio, C. , 2016, “ Composite 3D-Printed Metastructures for Low-Frequency and Broadband Vibration Absorption,” Proc. Natl. Acad. Sci. USA, 113(30), pp. 8386–8390.
Thomson, W. , 1887, “ On the Division of Space With Minimum Partitional Area,” London, Edinburgh, Dublin Philos. Mag. J. Sci., 24(151), pp. 503–514. [CrossRef]
Weaire, D. , and Phelan, R. , 1994, “ A Counter-Example to Kelvin's Conjecture on Minimal Surfaces,” Philos. Mag. Lett., 69(2), pp. 107–110. [CrossRef]
Gong, L. , Kyriakides, S. , and Jang, W. Y. , 2005, “ Compressive Response of Open-Cell Foams. Part I: Morphology and Elastic Properties,” Int. J. Solids Struct., 42(5), pp. 1355–1379. [CrossRef]
Gong, L. , and Kyriakides, S. , 2005, “ Compressive Response of Open Cell Foams—Part II: Initiation and Evolution of Crushing,” Int. J. Solids Struct., 42(5), pp. 1381–1399. [CrossRef]
Jang, W. Y. , and Kyriakides, S. , 2009, “ On the Crushing of Aluminum Open-Cell Foams: Part II Analysis,” Int. J. Solids Struct., 46(3), pp. 635–650. [CrossRef]
Kittel, C. , 2005, Introduction to Solid State Physics, 8th ed., Wiley, Hoboken, NJ.
Åberg, M. , and Gudmundson, P. , 1997, “ The Usage of Standard Finite Element Codes for Computation of Dispersion Relations in Materials With Periodic Microstructure,” J. Acoust. Soc. Am., 102(4), pp. 2007–2013. [CrossRef]
Wang, P. , Casadei, F. , Kang, S. H. , and Bertoldi, K. , 2015, “ Locally Resonant Band Gaps in Periodic Beam Lattices by Tuning Connectivity,” Phys. Rev. B, 91(2), p. 020103(R). [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic representation of (a) Kelvin foam, (b) simple-cubic, (c) octet, and (d) framed-cubic lattices and corresponding unit-cells in canonical form (lower left cells in Fig. 1(a)1(d)). In right column are the unit-cells considered for wave propagation analysis. (e) Reciprocal lattice unit-cell and corresponding IBZ for a cubic lattice.

Grahic Jump Location
Fig. 2

(a) Band diagram, (b) density of states, and (c) the first three mode shapes at high symmetry points of the IBZ for the Kelvin foam with relative density ρr = 15%

Grahic Jump Location
Fig. 3

Iso-frequency contour plots for the first three modes of Kelvin unit-cell with ρr = 15% in ky − kz plane at different levels of the kx component of the wave vector. Different colormaps are assigned for each mode to better illustrate the discrepancy among the modes.

Grahic Jump Location
Fig. 4

Normalized (a) phase and (b) group velocity of Kelvin unit cell with ρr = 15% for the first three modes at different frequencies that correspond to the kx = 0 iso-frequency contours in Fig. 3. The diamond, square, and circle markers represent first, second, and third modes, respectively. (In online version, the red diamond, green square, and blue circle markers represent first, second, and third modes, respectively.)

Grahic Jump Location
Fig. 5

Four-dimensional group velocity plots of the Kelvin foam with density ρr = 15%, representing preferential directions of wave propagation for the first three modes

Grahic Jump Location
Fig. 6

(a) Band diagram, (b) density of states, and (c) the first three mode shapes at high symmetry points of the IBZ for the simple-cubic lattice with relative density ρr = 15%

Grahic Jump Location
Fig. 7

Iso-frequency contour plots for the first three modes of the simple-cubic lattice with ρr = 15% in ky − kz plane at different levels of the kx component of the wave vector

Grahic Jump Location
Fig. 8

Normalized (a) phase and (b) group velocity of the simple-cubic lattice with ρr = 15% for the first three modes at different frequencies that correspond to the kx = 0 iso-frequency contours in Fig. 7. The diamond, square, and circle markers represent first, second, and third modes, respectively. (In online version, the red diamond, green square, and blue circle markers represent first, second, and third modes, respectively.)

Grahic Jump Location
Fig. 9

Four-dimensional group velocity plots of the simple-cubic lattice with density ρr = 15%, representing preferential directions of wave propagation for the first three modes

Grahic Jump Location
Fig. 10

(a) Band diagram, (b) density of states, and (c) the first three mode shapes at high symmetry points of the IBZ for the octet lattice with relative density ρr = 15%

Grahic Jump Location
Fig. 11

Iso-frequency contour plots for the first three modes of the octet lattice with ρr = 15% in ky − kz plane at different levels of the kx component of the wave vector

Grahic Jump Location
Fig. 12

Normalized (a) phase and (b) group velocity of the octet lattice with ρr = 15% for the first three modes at different frequencies that correspond to the kx = 0 iso-frequency contours in Fig. 11. The diamond, square, and circle markers represent first, second, and third modes, respectively. (In online version, the red diamond, green square, and blue circle markers represent first, second, and third modes, respectively.)

Grahic Jump Location
Fig. 13

Four-dimensional group velocity plots of the simple-cubic lattice with density ρr = 15%, representing preferential directions of wave propagation for the first three modes

Grahic Jump Location
Fig. 14

(a) Band diagram and (b) density of states for the framed-cubic lattice with relative density ρr = 15%. The shaded area represents the bandgap.

Grahic Jump Location
Fig. 15

Iso-frequency contour plots for the first three modes of the anisotropic Kelvin foam with ρr = 15% in ky − kz plane at different levels of the kx component of the wave vector

Grahic Jump Location
Fig. 16

Four-dimensional group velocity plots of the anisotropic Kelvin foam with density ρr = 15%, representing preferential directions of wave propagation for the first three modes

Grahic Jump Location
Fig. 17

Anisotropy index versus normalized frequency plots that correspond to the first three propagating modes for (a) Kelvin foam with ρr = 2%, (b) Kelvin foam with ρr = 15%, (c) Anisotropic Kelvin foam with ρr = 15%, (d) simple-cubic lattice with ρr = 2%, (e) simple-cubic lattice with ρr = 15%, (f) anisotropic cubic lattice with ρr = 15%, (g) octet lattice with ρr = 2%, (h) octet lattice with ρr = 15%, and (i) anisotropic octet lattice with ρr = 15%

Grahic Jump Location
Fig. 18

Transient numerical simulation snapshots for a single-point harmonic perturbation at Ω = 0.7 on a 10 × 10 × 10 Kelvin lattice at (a) 0.1 × 10−2, (b) 6.5 × 10−2, and (c) 8.8 × 10−2 s

Grahic Jump Location
Fig. 19

Transient numerical simulation snapshots for a single-point harmonic perturbation atΩ = 0.4 on a 20 × 20 × 20 simple-cubic lattice at (a) 0.3 × 10−2, (b) 2.1 × 10−2, and (c) 3.6 × 10−2 s

Grahic Jump Location
Fig. 20

Transient numerical simulation snapshots for a single-point harmonic perturbation at Ω = 7 on a 10 × 10 × 10 octet lattice at (a) 0.2 × 10−2, (b)1.6 × 10−2, and (c) 4.2 × 10−2 s

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