We consider an elastic triangular lattice (TL) containing tilted rotational resonators, as the one represented in Figure 1A. Point-wise masses m (black full circles) are considered at the triangular lattice nodes in Figure 1A, the lattice vectors being (1)t1=10L    and    t2=13L2, where L is the distance between nearest neighbors. Figure 1B shows the first Brillouin zone of the TL, together with its irreducible part (gray area). The high-symmetry points are (2)Γ=00,     M=2π3L1∕31   and   X=2π3L01.

The nodal points of the lattice whose mass is m are linked to each other by non-flexible, massless, extensible rods (thin lines) of longitudinal stiffness cℓ. The unit cell of the lattice (semitransparent yellow region in Figure 1A) contains a resonator, an equilateral triangle of side ℓ with point masses m_o attached to its vertices (empty circles in Figure 1A). The vertices of the resonators are linked to the nodal points of the TL by non-flexible, extensible rods of longitudinal stiffness cℓo (medium-thickness black lines in Figure 1A). In this paper, the resonators are assumed to be rigid, i.e., the longitudinal stiffness c_o of the links connecting the vertices of the resonators is such that co∕cℓo→+∞ and co∕cℓ→+∞. The resonators are tilted with respect to the external triangular lattice by an angle ϑ₀, marked in Figure 1A.

We now give some geometric definitions useful to represent the dispersion equation for the triangular lattice with resonators. We denote by b˜i, i = {1, 2, 3}, the position vector of the i^th mass relative to the center of mass r˜cm=L∕2(1,1∕3)T, where “T” denotes transposition. The explicit expression is (3)b˜i=bR^iβ˜1=bR^isinϑ0cosϑ0,     with     R^i=R^ϑϑ=2π(i−1)∕3,i={1,2,3}, where ϑ₀ is the tilting angle, b=ℓ∕3, and (4)R^ϑ=cosϑsinϑ−sinϑcosϑ, is the clockwise rotation matrix. The vector linking the triangular lattice to the i^th mass of the resonator in the reference cell n = 0 is (5)α˜i=R^iα˜1,i={1,2,3},    withα˜1=t2−r˜cm−b˜1=bsinϑ0−(B−bcosϑ0), where B=L∕3, b has been introduced in equation (3) and the matrix R^i is given in equation (3). Given the set of vectors (1) and (5), we introduce the corresponding projector matrices (6)τ^1=1L2t1t1T,      τ^2=1L2t2t2T,      τ^3=1L2t1−t2t1−t2T,Π^i=1ℓr2α˜iα˜iT,i={1,2,3},withℓr=||α˜i||=13L2+ℓ2−2ℓLcos(ϑ0).

The notation vu^T in equation (6) is used to denote the dyadic product v ⊗ u of two vectors u and v.

We consider time-harmonic elastic Bloch-Floquet waves propagating through the lattice. Following Tallarico et al. (2017), the Bloch-Floquet displacement wave’s amplitude with Bloch vector k is (7)Uk=u0T(k),ucmT(k),ϑ(k)T, where the vectors quantities u0T(k) and ucmT(k) are the in-plane displacements of the TL nodal points and of the center of mass of the resonators, respectively. In equation (7), ϑ(k) represents the angular displacement with respect to the equilibrium ϑ₀. In the time-harmonic regime, the equations of motion in the lattice characterized by the displacement (7) have the matrix form (8)Σ^k−ω2M^Uk=0, where ω is the Bloch-Floquet radian frequency and the vector U_k is given in equation (7). The inertia matrix that appears in equation (8) is (9)M^=diag(m,m,M,M,I), where M = 3m_o is the total mass of the resonator, I=moℓ2 is its moment of inertia, and m is the mass of the nodal points of the triangular lattice. In Tallarico et al. (2017), it has been shown that the stiffness matrix in equation (8) is (10)Σ^k=Σ^0,0(k)Σ^0,cm(k)Σ0,ϑ(k)Σ^0,cm†(k)Σ^cm,cmΣcm,ϑΣ0,ϑ†(k)Σcm,ϑ†Σϑ,ϑ=∑i=13−2cℓ(cos(k⋅ti)−1)τ^i−cℓoφi(k)Π^i−cℓoφi(k)Π^iR^i′b˜1+cℓoΠ^i−cℓoφi∗(k)Π^icℓoΠ^icℓoΠ^iR^i′b˜1−cℓo(φi(k)Π^iR^i′b˜1)†cℓo(Π^iR^i′b˜1)†−cℓob˜1T⋅(R^i′Π^iR^i′b˜1), where φ₁(k) = exp(−k ⋅ t₂), φ₂(k) = exp(−k ⋅ t₁), φ₃(k) = 1, and R^i′=d∕dϑR^ϑϑ=2π(i−1)∕3. Consider the 3 × 3 block independent of k that appears in equation (10). We observe that (11)σ=Σ^cm,cmΣcm,ϑΣcm,ϑ†Σϑ,ϑ=3cℓo∕2Î2×200Tcℓoℓ2sin2ϑ0∕(1+ℓ2∕L2−2ℓ∕Lcos(ϑ0)), where Î2×2 is the 2 × 2 identity matrix. The diagonal matrix (11) is the stiffness matrix for a single resonator for which the natural frequencies squared are (Tallarico et al., 2017) (12)Ωcm=32cℓoM,  and   Ωϑ=cℓoℓ2Isin2ϑ01+ℓ2∕L2−2ℓ∕Lcosϑ0.

In equation (12), Ω_cm is the frequency of oscillation of the center of mass of a single resonator, whereas the frequency Ω_ϑ describes the harmonic rotation of the resonator.

The elastic Bloch-Floquet waves in the doubly periodic structure of tilted resonators have interesting dispersion properties shown in Figure 2. A special feature is the Dirac-like cone with the vertex corresponding to k = 0, which is the main focus of this paragraph. Seeking non-trivial solutions for equation (8) requires (13)D(k,ω)=detΣ^k−ω2M^=0, whose roots ω vs k determine the dispersion of Bloch waves (see, e.g., Figure 2). At k = 0, the roots of the fifth-degree in Ω = ω² polynomial equation (13) can be found in their closed forms. Introducing the notation ΩΓ(i)=Ωk(i)k=0, with i the index of the root, we find (14)ΩΓ(1)=0,   ΩΓ(2)=Ωcm1+3mom   and   ΩΓ(3)=Ωϑ. where Ω_cm and Ω_ϑ have been introduced in equation (12). The first and second eigenvalues in equation (14) have multiplicity two, and the third one has multiplicity one. The geometric conditions (15)0<ℓL<12         and         |ϑ0|<ϑmax≡arcosℓL, guarantee that the trusses do not cross each another. We observe that it is possible to obtain a triple eigenvalue corresponding to ΩΓ(2)=ΩΓ(3), if there exists (16)m¯=cos2ϑ0+ℓ¯2−2ℓ¯cosϑ02ℓ¯cosϑ0−ℓ¯2−1>0, with m¯=3mo∕m and ℓ¯=ℓ∕L. We observe that (17)m¯>0⇔   cosϑ0−|sinϑ0|<ℓ¯<cosϑ0+|sinϑ0|.

The substitution of the expression (16) for m_o into the Bloch frequencies at Γ in equation (14) gives the frequency squared for the triple eigenvalue (18)ΩΓ(te)=−3cℓomsin2ϑ0ℓ¯2−2ℓ¯cosϑ0+cos2ϑ0, which is a positive quantity if the condition on ℓ¯ and ϑ₀ of equation (17) is satisfied.

Figure 3A represents the frequency dispersion surfaces for a TLR as a function of a set of Bloch wave vectors that comprise the first Brillouin zone (see Figure 1B). The lattice parameters have been chosen in such a way that equation (16) is satisfied. This implies the occurrence of a triple eigenvalue at Γ, as it can be seen by direct inspection of the optical part of the dispersion diagram. Specifically, we choose ℓ¯=0.21 and ϑ₀ = 0.82, which gives m¯=0.41. Moreover, we fix L=cℓ=1 and m = 0.8, which influences the maximum frequency of the acoustic modes. Finally, the choice cℓo=1.53 guarantees that the frequency of the triple eigenvalue (18) is (19)ΩΓ(te)=π.

Figures 3B,C show the slowness contours of Figure 3A around the triple eigenvalue’s frequency ω = π. Figures 3B,C refers to frequencies just above and just below ω = π, respectively. Figures 3B,C show that the dispersion in the vicinity of the triple eigenvalue is isotropic. In Figure 3D, we compare along the path MΓXM the optical branches of three different TLRs whose lattice parameters are listed in Table 1. The black solid line refers to set 1 in Table 1 which has been already used in Figure 3A. Hence, the dispersion around the triple eigenvalue’s frequency ω = π is linear, suggesting that the triple eigenvalue is a Dirac-like point. Other choices of the parameters are possible resulting in different effective group velocities at Γ. In Figure 3D, we use set 2 (red dashed line) and set 3 (blue dotted line) listed in Table 1. The chosen sets of parameters satisfy (19), which corresponds to the occurrence of a triple eigenvalue at Γ and ω = π. We observe that Dirac-like dispersion is robust over the chosen sets of the lattice parameters.

Figures 4A–C show the harmonic responses of a cluster with lattice parameters as in set 1 of Table 1. In these computations, three different linearly polarized forces have been used, each of which is oriented at 0, π/3, and π/6 with respect to the horizontal axis (see black arrows). In Figures 4A–C, we observe a localization pattern consistent with the flat band intersecting the Dirac cone at the triple eigenvalue. The symmetry axis of the localization pattern follows the polarization angle of the force. Figures 4D–F show the harmonic responses of a special cluster of resonators in which an inhomogeneity has been introduced via the tilting angle. The remaining parameters are listed in “set 1” of Table 1 and the harmonic force is the same as in Figures 4A–C. Above the thin black line, the resonators are tilted in the anticlockwise direction (ϑ₀ = − 0.82), while below the line, a clockwise tilting (ϑ₀ = 0.82) is implemented. This inhomogeneity introduces an interface that runs along the thin black line. It shall be pointed out that the dispersion surfaces of the lattice of resonators with clockwise and anticlockwise tilting are identical. In particular, the effective group velocities at the Dirac-like point are identical. Nevertheless, the harmonic response of the non-homogeneous cluster differs significantly from the corresponding responses of the homogeneously tilted cluster. In fact, we observe that a point force of frequency ω = π rad/s, corresponding to the Dirac-like point, triggers an edge wave traveling along the interface. The amplitude of the edge wave depends on the orientation of the harmonic point force, being larger for larger deflections from the horizontal direction (cf. Figures 4D–F). In each of the three panels, the elastic edge wave propagating along the interface has elliptic polarization whose principal axis is oriented at π/3 with respect to the interface. When the linear polarization angle of the source matches π/3 (see Figure 4F), the amplitude of the edge wave is greater than the other two cases for geometrical reasons.

In the same spirit as in Figure 4, Figure 5 shows the harmonic responses of clusters whose lattice parameters are listed in “set 2” (panels (A,B)) and “set 3” (panels (C,D)) of Table 1. The aim here is to illustrate how different dispersive properties near the Dirac-like point, already highlighted in Figure 3D, affect the harmonic responses of homogeneously tilted clusters (Figures 5A,C) and non-homogeneously tilted clusters (Figures 5B,D). The non-homogeneity considered here has the same meaning as in Figure 4. Figures 5A,C show localized patterns similar to that encountered in Figure 4A. Figures 5B,D show an edge wave traveling across the interface. We remark that the wavelength of the edge waves is larger for smaller effective group velocities at the Dirac-like point ω = π rad/s. This suggests that the dynamics of the edge waves is controlled by the effective group velocities at the Dirac-like point.

Figure 6 shows the modulus of the displacement field for a forced TLR containing a defect, which consists of a missing line of resonators, as shown in the magnified inset highlighted in yellow on the right of the figure. The lattice parameters used in this computation are listed in set 1 of Table 1 and the tilting angle is anticlockwise and clockwise, above and below the defect, respectively. The harmonic force is identical to the one used in Figure 4A and is exerted on a triangular lattice nodal point below the line defect (see blue arrow in the inset). We observe that the defect acts as a wave guide for an edge wave whose wavelength differs from the one in Figure 4B. We emphasize again that the wave-guiding behavior in Figure 6 differs significantly from the localization pattern in Figure 4A, the bulk homogeneous counterpart.

In Figure 8, we compare the modulus of the displacement field resulting from the interaction of an elastic shear wave with a cluster of resonators (Figure 8A) and with a cluster of resonators containing a crack (Figure 8B). The source of the excitation is a plane wave of frequency ω = π rad/s, which corresponds to the Dirac-like point for the periodic TLR (see Figure 3). Figure 8A shows that scattering of elastic waves is highly anisotropic, the displacement field being concentrated on the right side of the cluster. It is worthwhile noting that if the resonators are rotated in the anticlockwise direction (ϑ₀ = − 0.82), the displacement field is mirror-symmetric compared to the one in Figure 8A. The introduction of a crack within the cluster (Figure 8B) triggers the propagation of elastic waves around the crack itself. The displacement field and the corresponding stresses are still visibly concentrated around the right tip of the crack. This suggests that a coating of resonators in the Dirac-like regime is likely to lead to a left–right asymmetry in the propagation of the crack.

In Figure 9, long strips of resonators containing a crack interact with a shear plane wave impinging on the strip from above. Several arrangements for the resonators are considered. In Figure 9A,B, the resonators in the strip are homogeneously tilted in the clockwise (anticlockwise) direction. Similar to Figure 8A, this leads to an enhancement of the displacement field close to the tips of the cracks. Moreover, the results are mirror-symmetric about the vertical line passing through the center of the crack. This is consistent with what we observe in Figure 8B. In Figure 9C, the homogeneously tilted strip analyzed in Figure 9A has been replaced by a strip with an interface. The interface is represented by a line of missing resonators. The stiffness of the triangular lattice links that define the interface is assumed to be c_TL = 50 N/m, as in the exterior triangular lattice. In Figure 9D, the strip is similar to the one in Figure 9C, but anticlockwise tilting above the line and clockwise tilting below the line are implemented. In Figures 9C,D, the displacement field is mirror-symmetric with respect to a vertical line passing through the crack.

In Figure 10, a shear plane wave coming from above impinges at normal incidence on a cluster of resonators (Figure 10A) and on clusters of resonators containing a crack (Figure 10B–D). The frequency of the excitation is ω = 2.4 rad/s corresponding to the band gap in Figure 3A. It is remarked that the coating is not penetrated by the incident wave. In particular, Figure 10B shows that the structured cluster acts as a protective layer for the crack, as one would expect from the analysis of the dispersion diagram for Bloch waves. In Figures 10C,D, we introduce a defect consisting of a missing line of resonators along the extension of the crack. In Figure 10C, the tilting angle is homogeneous, whereas in Figure 10D, the resonators are tilted in opposite directions above and below the line defect. The stiffness of the links of the line defects is the same as of the exterior triangular lattice. Figures 10C,D show a displacement enhancement at the perimeter of the cluster, however away from the crack tip. Figure 10E highlights an edge wave traveling along the boundary of the cluster.

In Figures 11A–C, the angular frequency ω = 2.1 rad/s of the plane wave corresponds to the lower edge of the band gap of Figure 3A. In Figures 11D–F, the frequency ω = 2.7 rad/s corresponds to the upper edge of the band gap. For the lower edge frequency, although the cluster is partially protective (see Figure 11B), the introduction of the one-dimensional defect increases the stress concentration around the crack (Figure 11C), compared to the uncoated configuration (Figure 11A). A similar effect is reported for the upper edge of the band gap in Figures 11E,F. In the vicinity of the band gap edges, the coating of resonators enhances the displacement field around the crack, increasing the chances for the crack to propagate.