Wave propagation properties of one-dimensional acoustic metamaterials with nonlinear diatomic microstructure

Abstract

Acoustic metamaterials are artificial microstructured media, typically characterized by a periodic locally resonant cell. The cellular microstructure can be functionally customized to govern the propagation of elastic waves. A one-dimensional diatomic lattice with cubic inter-atomic coupling—described by a Lagrangian model—is assumed as minimal mechanical system simulating the essential undamped dynamics of nonlinear acoustic metamaterials. The linear dispersion properties are analytically determined by solving the linearized eigenproblem governing the free wave propagation in the small-amplitude oscillation range. The dispersion spectrum is composed by a low-frequency acoustic branch and a high-frequency optical branch. The two frequency branches are systematically separated by a stop band, whose amplitude is analytically derived. Superharmonic 3:1 internal resonances can occur within a wavenumber-dependent locus defined in the mechanical parameter space. A general asymptotic approach, based on the multiple scale method, is employed to determine the nonlinear dispersion properties. Accordingly, the nonlinear frequencies and waveforms are obtained for the two fundamental cases of non-resonant and superharmonically 3:1 resonant or nearly resonant lattices. Moreover, the invariant manifolds associated with the nonlinear waveforms are parametrically determined in the space of the two principal coordinates. Finally, some examples of non-resonant and resonant lattices are selected to discuss their nonlinear dispersion properties from a qualitative and quantitative viewpoint.

Appendices

Appendix: Model formulation

1.1 Hamilton’s principle

The kinetic energy for the one-dimensional lattice characterized by the diatomic periodic cell reads

$$\begin{aligned} {\mathcal {K}}=\tfrac{1}{2}M({\dot{U}})^2+\tfrac{1}{2}M_r({\dot{V}})^2 \end{aligned}$$

(67)

whereas the elastic potential energy takes the form

$$\begin{aligned} {\mathcal {U}}=\tfrac{1}{2}K(U-U_2)^2+\tfrac{1}{2}K(U_3-U)^2+K_r\varepsilon _r^2 \end{aligned}$$

(68)

where—according to a second-order kinematics—the secondary spring extension reads

$$\begin{aligned} \varepsilon _r=\varepsilon ^\circ _r+\frac{(V-U)^2}{2R} \end{aligned}$$

(69)

and $\varepsilon ^\circ _r=H/K_r$ is the spring preextension responsible for the spring pretension.

Introducing the work done by the external forces ${\mathcal {W}}=F_2U_2+F_3U_3$, the Hamiltonian action between two instants of time $t_1$ and $t_2$ takes the integral expression

$$\begin{aligned} {\mathcal {H}}=\int _{t_1}^{t_2}\left( {\mathcal {K}}-{\mathcal {V}}\right) \hbox {d}t \end{aligned}$$

(70)

where ${\mathcal {V}}={\mathcal {U}}-{\mathcal {W}}$ is the total potential energy. According to the Hamilton’s Principle, the Hamiltonian action can be imposed to be stationary to obtain the nonlinear equations of motion (3)-(4) and (5), where nondimensional variables and parameters have been introduced.

1.2 Matrices and coefficients of the Lagrangian model

The 2-by-2 mass and stiffness matrices governing the equation of motion (10) are

$$\begin{aligned} {\mathbf {M}}= \left[ \begin{array}{cc} 1+\varrho ^2 &{} \varrho ^2\\ \varrho ^2 &{} \varrho ^2 \end{array}\right] ,\quad {\mathbf {K}}= \left[ \begin{array}{cc} 1-\cos \beta &{} 0\\ 0 &{} 2\mu \end{array}\right] \end{aligned}$$

(71)

whereas the 2-by-1 operator accounting for the cubic nonlinearities reads

$$\begin{aligned} {\mathbf {n}}({\mathbf {u}},{\mathbf {u}},{\mathbf {u}})=\left( \begin{array}{c} 0 \\ \eta \, w^3 \end{array}\right) \end{aligned}$$

(72)

The transformed 2-by-1 operator accounting for the cubic nonlinearities reads

$$\begin{aligned} {\mathbf {c}}({\mathbf {q}},{\mathbf {q}},{\mathbf {q}})=-\eta \,{\varvec{\varphi }}_2\left( {\varvec{\varphi }}_2^\top {\mathbf {q}}\right) \left( {\varvec{\varphi }}_2^\top {\mathbf {q}}\right) \left( {\varvec{\varphi }}_2^\top {\mathbf {q}}\right) \end{aligned}$$

(73)

where ${\varvec{\varphi }}_2=\left( {\varphi }^-_2,{\varphi }^+_2\right)$ is the vector collecting the second components of the mass-normalized waveforms

$$\begin{aligned} {\varphi }^-_2=\frac{2}{D^-_2},\qquad {\varphi }^+_2=\frac{2}{D^+_2} \end{aligned}$$

(74)

where the denominators are

$$\begin{aligned} D^\mp _2\!=&\!\Big [1\!+\!\varrho ^2-2\big (1-\varrho ^2\big )I_2^\mp (\beta )+\nonumber \\&+\,\big (1\!+\!\varrho ^2\big )(I_2^\mp (\beta ))^2\Big ]^{\!\tfrac{1}{2}} \end{aligned}$$

(75)

and the $\beta$-dependent auxiliary parameters

$$\begin{aligned} I^\mp _2(\beta )=\frac{2\mu \big (1+\varrho ^2\big )\pm S(\beta )}{\varrho ^2(1-\cos \beta )} \end{aligned}$$

(76)

The components of the nonlinearity vector ${\mathbf {c}}({\mathbf {q}},{\mathbf {q}},{\mathbf {q}})=\left( {c}^-({\mathbf {q}},{\mathbf {q}},{\mathbf {q}}),{c}^+({\mathbf {q}},{\mathbf {q}},{\mathbf {q}})\right)$ are complete cubic polinomials

$$\begin{aligned}&{c}^-({\mathbf {q}},{\mathbf {q}},{\mathbf {q}})=c_{111}(q^-)^3+c_{112}(q^-)^2q^++\\&\qquad \qquad \qquad +\,c_{122}q^-(q^+)^2+c_{222}(q^+)^3\nonumber \\&{c}^+({\mathbf {q}},{\mathbf {q}},{\mathbf {q}})=d_{111}(q^-)^3+d_{112}(q^-)^2q^++\nonumber \\&\qquad \qquad \qquad +\,d_{122}q^-(q^+)^2+d_{222}(q^+)^3\nonumber \end{aligned}$$

(77)

where the coefficients are

$$\begin{aligned}&c_{111}=-\eta (\varphi ^-_2)^4,&c_{112}=-3\eta (\varphi ^-_2)^3(\varphi ^+_2)\\&c_{122}=-3\eta (\varphi ^-_2)^2(\varphi ^+_2)^2,&c_{222}=-\eta (\varphi ^-_2)(\varphi ^+_2)^3\nonumber \\&d_{111}=-\eta (\varphi ^-_2)^3(\varphi ^+_2),&d_{112}=-3\eta (\varphi ^-_2)^2(\varphi ^+_2)^2\nonumber \\&d_{122}=-3\eta (\varphi ^-_2)(\varphi ^+_2)^3,&d_{222}=-\eta (\varphi ^+_2)^4\nonumber \end{aligned}$$

(78)

1.3 Maxima of the resonant locus

The analytical expressions for the $\beta$-dependent maxima of the parametric locus $\mathcal {S}$ portrayed in Fig. 4 are

$\bullet$:

for the $\varrho ^2$-maxima

$$\begin{aligned} \varrho ^2&=\frac{16}{15129}\!\left( 400\sqrt{2(1-\cos \beta )}\csc \frac{\beta }{2}+881\!\right) \\ \mu&=\frac{8}{25}(1-\cos \beta )\nonumber \end{aligned}$$

(79)

$\bullet$:

for the $\mu$-maxima

$$\begin{aligned} \varrho ^2&=\frac{16}{25}\\ \mu&=\frac{8}{9}(1-\cos \beta )\nonumber \end{aligned}$$

(80)

and for $\beta =\pi$ correspond to the mechanical parameter combinations $(\varrho ^2_\mathrm {max},\mu )=(16/9,16/25)$ and $(\varrho ^2,\mu _\mathrm {max})=(16/25,16/9)$ respectively.

Appendix: Alternative approach

The asymptotic strategy proposed in Sect. 2 requires the enforcement of the quasi-periodicity conditions on the nonlinear equations of motion (3), (4) governing the free vibration dynamics of the diatomic cell. Differently, the common procedure requires to expand the nonlinear equations according to the multiple scale method and, subsequently, to enforce the quasi-periodicity conditions in the linear equations governing each order of the asymptotic expansion [15, 37]. In order to demonstrate the substantial equivalence of the two approaches, the latter procedure can be demonstrated to originate the same formal hierarchy of differential linear problems. To this purpose, the nonlinear equations (3)–(5) can be partitioned in the form

$$\begin{aligned} \left[ {\begin{array}{cc} {\mathbf{M}}&{}{\mathbf{O}}\\ {\mathbf{O}}&{}{\mathbf{O}} \end{array}} \right] \!\! \left( \!{\begin{array}{c} {\ddot{\mathbf{u}}_a}\\ {\ddot{\mathbf{u}}_p} \end{array}}\!\right) \! +\!\left[ {\begin{array}{cc} {{{\mathbf{K}}_{aa}^{n}}}&{}{{{\mathbf{K}}_{ap}}}\\ {{{\mathbf{K}}_{pa}}}&{}{{{\mathbf{K}}_{pp}}} \end{array}} \right] \!\! \left( \!{\begin{array}{c} {{{\mathbf{u}}_a}}\\ {{{\mathbf{u}}_p}} \end{array}}\!\right) \!=\! \left( \!{\begin{array}{c} {{{\mathbf{0}}}}\\ {{{\mathbf{f}}_p}} \end{array}}\!\right) \end{aligned}$$

(81)

where the active displacement subvector ${\mathbf{u}}_a=(u,w)$ has been distinguished from the passive displacement subvector ${\mathbf{u}}_p=(u_2,u_3)$ and the passive force subvector ${\mathbf{f}}_p=(f_2,f_3)$. The submatrix ${\mathbf{K}}_{aa}^n={\mathbf{K}}_{aa}+{\mathbf{N}}({\mathbf{u}}_a)$, where ${\mathbf{N}}({\mathbf{u}}_a)$ accounts for the nonlinearities. Expanding all the variables in series of the small $\epsilon$-parameter

$$\begin{aligned} {{\mathbf{u}}_a}=&\,\epsilon {{\mathbf{u}}_{a1}}({T_0},{T_1},{T_2})+{\epsilon ^2}{{\mathbf{u}}_{a2}}({T_0},{T_1},{T_2})+\nonumber \\&+{\epsilon ^3}{{\mathbf{u}}_{a3}}({T_0},{T_1},{T_2}) + {\mathcal {O}}(\epsilon ^4)\nonumber \\ {{\mathbf{u}}_p}=&\,\epsilon {{\mathbf{u}}_{p1}}({T_0},{T_1},{T_2})+{\epsilon ^2}{{\mathbf{u}}_{p2}}({T_0},{T_1},{T_2})+\nonumber \\ {}&+{\epsilon ^3}{{\mathbf{u}}_{p3}}({T_0},{T_1},{T_2}) +\, {\mathcal {O}}(\epsilon ^4)\nonumber \\ {{\mathbf{f}}_p}=&\,\epsilon {{\mathbf{f}}_{p1}}({T_0},{T_1},{T_2})+{\epsilon ^2}{{\mathbf{f}}_{p2}}({T_0},{T_1},{T_2})+\nonumber \\ {}&+\,{\epsilon ^3}{{\mathbf{f}}_{p3}}({T_0},{T_1},{T_2}) + {\mathcal {O}}(\epsilon ^4) \end{aligned}$$

(82)

Introducing the expansion into the nonlinear differential equation (81) and equating all the terms of like $\epsilon$-powers yields a ordered hierarchy of linear differential equation

$\bullet$:

Order $\epsilon$

$$\begin{aligned} \left[ {\begin{array}{cc} {\mathbf{M}}&{}{\mathbf{O}}\\ {\mathbf{O}}&{}{\mathbf{O}} \end{array}} \right] \!\! \left( \!{\begin{array}{c} {D_0^2 {\mathbf{u}}_{a1}}\\ {D_0^2 {\mathbf{u}}_{p1}} \end{array}}\!\right) \! +\!\left[ {\begin{array}{cc} {{{\mathbf{K}}_{aa}}}&{}{{{\mathbf{K}}_{ap}}}\\ {{{\mathbf{K}}_{pa}}}&{}{{{\mathbf{K}}_{pp}}} \end{array}} \right] \!\! \left( \!{\begin{array}{c} {{{\mathbf{u}}_{a1}}}\\ {{{\mathbf{u}}_{p1}}} \end{array}}\!\right) \!=\! \left( \!{\begin{array}{c} {{{\mathbf{0}}}}\\ {{{\mathbf{f}}_{p1}}} \end{array}}\!\right) \end{aligned}$$

(83)

$\bullet$:

Order $\epsilon ^2$

$$\begin{aligned} \left[ {\begin{array}{cc} {\mathbf{M}}&{}{\mathbf{O}}\\ {\mathbf{O}}&{}{\mathbf{O}} \end{array}} \right] \!\! \left( \!{\begin{array}{c} {D_0^2 {\mathbf{u}}_{a2}}\\ {D_0^2 {\mathbf{u}}_{p2}} \end{array}}\!\right) \! +\!\left[ {\begin{array}{cc} {{{\mathbf{K}}_{aa}}}&{}{{{\mathbf{K}}_{ap}}}\\ {{{\mathbf{K}}_{pa}}}&{}{{{\mathbf{K}}_{pp}}} \end{array}} \right] \!\! \left( \!{\begin{array}{c} {{{\mathbf{u}}_{a2}}}\\ {{{\mathbf{u}}_{p2}}} \end{array}}\!\right) \!=\! \left( \!{\begin{array}{c} {{{\mathbf{f}}_{a2}}}\\ {{{\mathbf{f}}_{p2}}} \end{array}}\!\right) \end{aligned}$$

(84)

$\bullet$:

Order $\epsilon ^3$

$$\begin{aligned} \left[ {\begin{array}{cc} {\mathbf{M}}&{}{\mathbf{O}}\\ {\mathbf{O}}&{}{\mathbf{O}} \end{array}} \right] \!\! \left( \!{\begin{array}{c} {D_0^2 {\mathbf{u}}_{a3}}\\ {D_0^2 {\mathbf{u}}_{p3}} \end{array}}\!\right) \! +\!\left[ {\begin{array}{cc} {{{\mathbf{K}}_{aa}}}&{}{{{\mathbf{K}}_{ap}}}\\ {{{\mathbf{K}}_{pa}}}&{}{{{\mathbf{K}}_{pp}}} \end{array}} \right] \!\! \left( \!{\begin{array}{c} {{{\mathbf{u}}_{a3}}}\\ {{{\mathbf{u}}_{p3}}} \end{array}}\!\right) \!=\! \left( \!{\begin{array}{c} {{{\mathbf{f}}_{a3}}}\\ {{{\mathbf{f}}_{p3}}} \end{array}}\!\right) \end{aligned}$$

(85)

where the force vectors ${\mathbf{f}}_{a2}=-2{\mathbf{M}}D_0D_1{\mathbf{u}}_{a1}$ and ${\mathbf{f}}_{a3}=-2{\mathbf{M}}{D_0}{D_1}{{\mathbf{q}}_{a2}}-2{\mathbf{M}}{D_0}{D_2}{{\mathbf{q}}_{a1}}-{\mathbf{M}}D_1^2{{\mathbf{q}}_{a1}}+{{\mathbf{N}}_2}{{\mathbf{q}}_{a1}}$.

The linear differential problem (83) at the $\epsilon$-order can be reformulated by separating the dynamic (upper) part from the quasi-static (lower) part, yielding

$$\begin{aligned}&{\mathbf{M}}{D_0^2 {\mathbf{u}}_{a1}}+{\mathbf{K}}_{aa}{\mathbf{u}}_{a1}+{\mathbf{K}}_{ap}{\mathbf{u}}_{p1}={\mathbf{0}}\\&{\mathbf{K}}_{pa}{\mathbf{u}}_{a1}+{\mathbf{K}}_{pp}{\mathbf{u}}_{p1}={\mathbf{f}}_{p1}\nonumber \end{aligned}$$

(86)

where the quasi-periodicity conditions (6) can be applied on the passive displacement and force vectors. Consequently, employing the quasi-static condensation rules ${\mathbf{u}}_{p1}={\mathbf{L}}_{pa}{\mathbf{u}}_{a1}$ and ${\mathbf{f}}_{p1}=\left( {\mathbf{K}}_{pa}+{\mathbf{K}}_{pp}{\mathbf{L}}_{pa}\right) {\mathbf{u}}_{a1}$, the condensed homogeneous equation in the active displacement vector reads

$$\begin{aligned}&{\mathbf{M}}{D_0^2 {\mathbf{u}}_{a1}}+\left( {\mathbf{K}}_{aa}+{\mathbf{K}}_{ap}{\mathbf{L}}_{pa}\right) {\mathbf{u}}_{a1}={\mathbf{0}} \end{aligned}$$

(87)

where the auxiliary matrix

$$\begin{aligned}&{{\mathbf{L}}_{pa}}=B^{-1}\left( {\begin{array}{*{20}{c}} {{\mathbf{K}}_{pa}^{(2)}+{\mathbf{K}}_{pa}^{(1)}{\text {e}}^{-\imath \beta }}\\ {{\mathbf{K}}_{pa}^{(2)}{\text {e}}^{-\imath \beta }+{\mathbf{K}}_{pa}^{(1)}{\text {e}}^{-2\imath \beta }} \end{array}}\right) \end{aligned}$$

(88)

where ${\mathbf{K}}_{pa}^{(i)}$ is the generic row of the 2-by-2 matrix ${\mathbf{K}}_{pa}$ ($i=1,2$) and the scalar quantity

$$\begin{aligned}&B\!=\!-\!\left[ {K_{pp}^{(21)}+\left( K_{pp}^{(22)}+K_{pp}^{(11)}\right) {\text {e}}^{-\imath \beta }+K_{pp}^{(12)}{\text {e}}^{-2\imath \beta }}\right] \end{aligned}$$

(89)

where $K_{pp}^{(ij)}$ is the generic component of the 2-by-2 matrix ${\mathbf{K}}_{pp}$ ($i,j=1,2$). Since $({\mathbf{K}}_{aa}+{\mathbf{K}}_{ap}{\mathbf{L}}_{pa})$ can be verified to coincide with ${\mathbf{K}}(\beta )$, the linear equation (87) states a differential problem that formally coincides with the linear part of Eq. (10). Consequently, the same eigenpairs $(\lambda ,{\varvec{\phi }})$ satisfy the eigenproblems associated to both the equations. Thus, recalling the matrix ${\varvec{\Phi }}$ of the mass-normalized eigenvectors (${\varvec{\Phi }}^\top {\mathbf{M}}{\varvec{\Phi }}={\mathbf{I}}$) and introducing the change of variable ${\mathbf{u}}_{a1}={\varvec{\Phi }}{\mathbf{q}}_{1}$ from the active displacement vector ${\mathbf{u}}_{a1}$ to the principal coordinate vector ${\mathbf{q}}_{1}$, Eq. (87) becomes

$$\begin{aligned} D_0^2\,{\mathbf {q}}_1+{\varvec{\Lambda }}{\mathbf {q}}_1={\mathbf {0}} \end{aligned}$$

(90)

which exactly coincides with the $\epsilon$-order equation (17). Moreover, the passive displacements are related to the principal coordinates through the relation ${\mathbf{u}}_{p1}={\mathbf{L}}_{pa}{\varvec{\Phi }}{\mathbf{q}}_{1}$.

The homogeneous problem associated with the linear differential equation (84) at the $\epsilon ^2$-order is formally coincident with that associated with the linear differential equation (83). Therefore, the same eigenvector matrix ${\varvec{\Phi }}$ can conveniently be employed to perform the change of variables ${\mathbf{u}}_{a2}={\varvec{\Phi }}{\mathbf{q}}_{2}$, yielding

$$\begin{aligned} D_0^2\,{\mathbf {q}}_2+{\varvec{\Lambda }}{\mathbf {q}}_2=-2{\varvec{\Phi }}^\top {\mathbf{M}}{\varvec{\Phi }}D_0D_1{\mathbf {q}}_1 \end{aligned}$$

(91)

which, recalling the mass-normalization rule ${\varvec{\Phi }}^\top {\mathbf{M}}{\varvec{\Phi }}={\mathbf{I}}$, actually coincides with the $\epsilon ^2$-order equation (18).

The homogeneous problem associated with the linear differential equation (85) at the $\epsilon ^3$-order is again formally coincident with that associated with the linear differential equation (83). Therefore, the eigenvector matrix ${\varvec{\Phi }}$ can conveniently be employed to perform the change of variables ${\mathbf{u}}_{a3}={\varvec{\Phi }}{\mathbf{q}}_{3}$, yielding

$$\begin{aligned}&D_0^2\,{\mathbf {q}}_3+{\varvec{\Lambda }}{\mathbf {q}}_3=-2{\varvec{\Phi }}^\top \!{\mathbf{M}}{\varvec{\Phi }}D_0D_1{\mathbf {q}}_2+\\&\quad -\,2{\varvec{\Phi }}^\top {\mathbf{M}}{\varvec{\Phi }}D_0D_2{\mathbf {q}}_1-{\varvec{\Phi }}^\top \!{\mathbf{M}}{\varvec{\Phi }}D_1^2{\mathbf {q}}_1 -{\varvec{\Phi }}^\top {\mathbf{N}}_2{\varvec{\Phi }}{\mathbf {q}}_1\nonumber \end{aligned}$$

(92)

which, recalling the mass-normalization rule ${\varvec{\Phi }}^\top {\mathbf{M}}{\varvec{\Phi }}={\mathbf{I}}$ and verifying that ${\varvec{\Phi }}^\top {\mathbf{N}}_2{\varvec{\Phi }}{\mathbf {q}}_1=-{\mathbf{c}}({\mathbf {q}}_1,{\mathbf {q}}_1,{\mathbf {q}}_1)$, actually coincides with the $\epsilon ^3$-order equation (19).