Nonlinear wave propagation in locally dissipative metamaterials via Hamiltonian perturbation approach

Abstract

The cellular microstructure of periodic architected materials can be enriched by local intracellular mechanisms providing innovative distributed functionalities. Specifically, high-performing mechanical metamaterials can be realized by coupling the low-dissipative cellular microstructure with a periodic distribution of tunable damped oscillators, or resonators, vibrating at relatively high amplitudes. The benefit is the actual possibility of combining the design of wave-stopping bands with enhanced energy dissipation properties. This paper investigates the nonlinear dispersion properties of an archetypal mechanical metamaterial, represented by a one-dimensional lattice model characterized by a diatomic periodic cell. The intracellular interatomic interactions feature geometric and constitutive nonlinearities, which determine cubic coupling between the lattice and the resonators. The non-dissipative part of the coupling can be designed to exhibit a softening or a hardening behavior, by independently tuning the geometric and elastic stiffnesses. The nonlinear wavefrequencies and waveforms away from internal resonances are analytically determined by adopting a perturbation technique. The employed approach makes use of tools borrowed from Hamiltonian perturbation theory, together with techniques often used in the context of nearly-integrable Hamiltonian systems.The dispersion spectra are determined in closed, asymptotically approximate, form as a nonlinear function of the time-dependent decreasing amplitude decrement. The invariant manifolds defined by the harmonic periodic motions are also analytically determined. The asymptotic results are further validated numerically.

Appendix

1.1 Proof of Proposition 2.1

Let $P_{\lambda }({\varvec{A}})$ denote the characteristic polynomial of ${\varvec{A}}$ expressed as

$$\begin{aligned} P_{\lambda }({\varvec{A}})= & {} \lambda ^4+\xi \chi \lambda ^3+2 p \lambda ^2 + 2 (\chi -1)\xi \theta \lambda +q \nonumber \\= & {} \lambda ^2 \left[ \lambda ^2+ \xi \chi \lambda + p \right] \nonumber \\&+ \left[ p \lambda ^2 + 2 (\chi -1)\xi \theta \lambda + q\right] \text{, } \end{aligned}$$

(45)

where $p:=\mu \chi +\theta$ and $q:=4 (\chi -1) \mu \theta$, respectively.

A sufficient condition for the statement to be true is that the two polynomials between square brackets do not possess real roots. This happens if either $\xi ^2 \chi ^2 < 4 p$ or $(\chi -1) \xi ^2 \theta < 4 \mu p$, respectively. It is immediate to check that both of them hold under condition (13).

The property $\alpha _{1,2}<0$ follows from the celebrated Routh-Hurwitz Theorem [82]. More precisely, from (45) it is possible to define

$$\begin{aligned} {\mathfrak {M}}:=\{{\mathfrak {m}}_{i,j}\}= \begin{pmatrix} \xi \chi &{} 1 &{} 0 &{} 0 \\ 2 (\chi -1)\xi \theta &{} 2p &{} \xi \chi &{} 1 \\ 0 &{} q &{} 2 (\chi -1)\xi \theta &{} 2p \\ 0 &{} 0 &{} 0 &{} q \end{pmatrix} \text{, } \end{aligned}$$

and the quantities $\varDelta _k:=\det \left( {\mathfrak {M}}^{( \le k)}\right)$, where ${\mathfrak {M}}^{( \le k)}:=\{{\mathfrak {m}}_{i,j}\}_{i,j \le k}$ is the square submatrix of ${\mathfrak {M}}$ formed by the first k rows and columns. The property $\Re \lambda _j <0$ for $\xi >0$ easily follows from the mentioned Theorem, by observing that either $\varDelta _1 \equiv \xi \chi$, or

$$\begin{aligned} \varDelta _2&=2 \xi ( \mu \chi ^2 + \theta ), \varDelta _3=4 \theta ^2 \xi ^2 (\chi -1), \\ \varDelta _4&=16 \mu \theta ^3 \xi ^2 (\chi -1)^2 \text{, } \end{aligned}$$

are positive by assumptions on the parameters.

Finally, the property $\beta _{1,2}>0$ is easily shown by reductio ad absurdum. For this purpose, it is worth recalling that the eigenvalues are non-purely real, as previously shown. On the other hand, Prop. 5.1 states that for any $\varvec{{\tilde{\zeta }}}:=({\tilde{\xi }},{\tilde{\varrho }},{\tilde{\mu }},{\tilde{\theta }})$ satisfying Eq. (36) and sufficiently small $\xi$, it turns out that $\beta _{1,2}=\beta _{1,2}(\varvec{{\tilde{\zeta }}})>0$. Next, suppose by contradiction, that for some other value $\varvec{{\hat{\zeta }}}$ (under the sole assumption (13)), one has $\beta _{1}(\varvec{{\hat{\zeta }}})<0$ (the argument for $\beta _2$ is the same). Hence, once defined, for all $j=1,...,4$, $\zeta _j(s_j):=s_j {\tilde{\zeta }}_j + (1-s_j) {\hat{\zeta }}_j$, with $\varvec{s} \in [0,1]^4$, there exists $\varvec{s}^*$ such that $\beta _1(\varvec{\zeta }(\varvec{s}^*))=0$, i.e., $\lambda _{1} \in {\mathbb {R}}$, which is a contradiction.

1.2 Proof of Prop 5.1

The stated range of variation of $\varTheta$ follows directly from (36) and the fact that $\chi >1$ by definition. By assumption, the characteristic polynomial $P_{\lambda }({\varvec{A}})$ is factorized as

$$\begin{aligned} P_{\lambda }({\varvec{A}})=\prod _{j=1}^{4}(\lambda -\lambda _j)\text{, } \end{aligned}$$

(46)

with $\lambda _{2k-1}={\bar{\lambda }}_{2k}$, $k=1,2$. By using the expansions (37) in (14), then substituting back into (46) and finally comparing the obtained expression with (45), one gets at zero order in $\xi$

$$\begin{aligned} b_1+b_2&= 2(\mu \chi + \theta ), \end{aligned}$$

(47)

$$\begin{aligned} b_1 b_2&= 4 \theta \mu (\chi -1), \end{aligned}$$

(48)

which give (39). On the other hand, at first order in $\xi$, the following set of conditions is obtained:

$$\begin{aligned} b_2 a_1 +b_1 a_2&= (\chi -1) \theta , \end{aligned}$$

(49)

$$\begin{aligned} 2(a_1+a_2)&= \chi , \end{aligned}$$

(50)

leading to (38).

Focusing first on the non-resonance property, it is clear from (16) that

$$\begin{aligned}&{\varvec{\varPsi }}={\varvec{\varPsi }}_0+O(\xi ),\nonumber \\&{\varvec{\varPsi }}_0:={\varvec{\varPsi }}|_{(\lambda _1,\lambda _2,\lambda _3, \lambda _4)=(\beta _1,\beta _1,\beta _2,\beta _2)} \text{. } \end{aligned}$$

(51)

In particular, the third row entries of ${\varvec{\varPsi }}_0$ are pure real at order zero in $\xi$. Hence, as $\xi _1=O(\xi )$ by definition, a comparison between (12) and (23) yields

$$\begin{aligned} \Im \gamma _{\varvec{\nu }}=O(\xi ) \text{. } \end{aligned}$$

(52)

Moreover, from (51), it is easy to check that

$$\begin{aligned} \det \left( {\varvec{\varPsi }}_0 \right) =&16 (\beta _1 \beta _2)^{-3} (\beta _{2}-\beta _{1})^2 \\&\times (\beta _{1}+\beta _2)^2 (\chi -1)\theta ^2 \text{, } \end{aligned}$$

i.e., ${\varvec{\varPsi }}_0$ is invertible away from ${\mathfrak {R}}$, so is ${\varvec{\varPsi }}$, for sufficiently small $\xi$. In particular, the fourth column of ${\varvec{\varPsi }}_0^{-1}$ yields

$$\begin{aligned} r_{1,2}&=\mp i [2(b_1-b_2)]^{-1}\sqrt{b_1} + O(\xi ) \text{, }\\ r_{3,4}&=\pm i [2(b_1-b_2)]^{-1}\sqrt{b_2} + O(\xi ) \text{, } \end{aligned}$$

where the expression in terms of the parameters is obtained from (39). The latter implies $\Re r_j=O(\xi )$, which compared with (52) gives (40).

Focusing finally on Eq. (41), it can be observed that $\lim _{(\xi ,\theta ) \rightarrow (0,0)^+} (\beta _{1},\beta _{2})=(0,2 \sqrt{\mu \chi })$. Hence, by continuity, for any sufficiently small $\xi$ and $\theta$, the curve $(\beta _{1}(\theta ),\beta _{2}(\theta ))$ “starts” inside the region

$$\begin{aligned} {\mathfrak {T}}:=\{\beta _2>3 \beta _1, \,\beta _1>0\} \subset {\mathbb {R}}^2 \text{. } \end{aligned}$$

The statement easily follows for sufficiently small $\xi$ by using (39). In fact, it is easy to check that, under assumption (36), one has $b_2>9 b_1$ for all $\theta \in (0,1)$.

1.3 Expression of the coefficients $\gamma _{\varvec{\nu }}$

Recalling that $\zeta :=\xi _1 \xi _2 \xi _3$ and $\kappa :=\mu -\eta$ and denoting by $\varPsi _{ij}$ the elements of matrix $\varvec{\varPsi }$, the coefficients $\gamma _{\varvec{\nu }}$ read

$$\begin{aligned} \gamma _{(3,0,0,0)}&=\varPsi _{21}^3 \kappa -\zeta \varPsi _{21}^2 \varPsi _{41} \chi +\zeta \varPsi _{21}^2 \varPsi _{31}^2\\ \gamma _{(0,3,0,0)}&=\varPsi _{22}^3 \kappa -\zeta \varPsi _{22}^2 \varPsi _{42} \chi +\zeta \varPsi _{22}^2 \varPsi _{32}^2\\ \gamma _{(0,0,3,0)}&= \varPsi _{23}^3 \kappa + \zeta \varPsi _{23}^3 (\varPsi _{33}-\chi \varPsi _{43})\\ \gamma _{(0,0,0,3)}&= \varPsi _{24}^3 \kappa + \zeta \varPsi _{24}^2 ( \varPsi _{34}-\varPsi _{44}\chi )\\ \gamma _{(1,1,1,0)}&= 6 \varPsi _{21} \varPsi _{22} \varPsi _{23} \kappa \\&- 2 \zeta \chi (\varPsi _{21} \varPsi _{22} \varPsi _{43} - \varPsi _{21} \varPsi _{23} \varPsi _{42} - \varPsi _{22} \varPsi _{23} \varPsi _{41})\\&+ 2 \zeta (\varPsi _{21}\varPsi _{22},\varPsi _{33}+\varPsi _{21}\varPsi _{23}\varPsi _{32} +\varPsi _{22}\varPsi _{23}\varPsi _{31})\\ \gamma _{(1,0,1,1)}&= 6 \varPsi _{21} \varPsi _{23} \varPsi _{24} \kappa \\&- 2 \zeta \chi (\varPsi _{21} \varPsi _{23} \varPsi _{44} - \varPsi _{21} \varPsi _{24} \varPsi _{43} - \varPsi _{23} \varPsi _{24} \varPsi _{41})\\&+ 2 \zeta (\varPsi _{21}\varPsi _{23}\varPsi _{34}+\varPsi _{21}\varPsi _{24}\varPsi _{33} +\varPsi _{23}\varPsi _{24}\varPsi _{31})\\ \gamma _{(0,1,1,1)}&= 6 \varPsi _{22} \varPsi _{23} \varPsi _{24} \kappa \\&- 2 \zeta \chi (\varPsi _{22} \varPsi _{23} \varPsi _{44} - \varPsi _{22} \varPsi _{24} \varPsi _{43} - \varPsi _{23} \varPsi _{24} \varPsi _{42})\\&+ 2 \zeta (\varPsi _{22}\varPsi _{23}\varPsi _{34}+\varPsi _{22}\varPsi _{24}\varPsi _{33} +\varPsi _{23}\varPsi _{24}\varPsi _{32})\\ \gamma _{(0,0,1,2)}&= 3 \varPsi _{23} \varPsi _{24}^2 \kappa - \zeta \chi (\varPsi _{24}^2 \varPsi _{43}-2 \varPsi _{23} \varPsi _{24} \varPsi _{44}) \\&+ \zeta (\varPsi _{24}^2 \varPsi _{33}+2 \varPsi _{23} \varPsi _{24} \varPsi _{34})\\ \gamma _{(0,1,0,2)}&= 3 \varPsi _{22} \varPsi _{24}^2 \kappa - \zeta \chi (\varPsi _{24}^2 \varPsi _{42}-2 \varPsi _{22} \varPsi _{24} \varPsi _{44}) \\&+ \zeta (\varPsi _{24}^2 \varPsi _{32}+2 \varPsi _{22} \varPsi _{24} \varPsi _{34})\\ \gamma _{(1,0,0,2)}&= 3 \varPsi _{21} \varPsi _{24}^2 \kappa -\zeta \chi (\varPsi _{24}^2 \varPsi _{41}-2 \varPsi _{21} \varPsi _{24} \varPsi _{44}) \\&+ \zeta (\varPsi _{24}^2 \varPsi _{31}+2 \varPsi _{21} \varPsi _{24} \varPsi _{34})\\ \gamma _{(0,0,2,1)}&= 3 \varPsi _{23}^2 \varPsi _{24} \kappa - \zeta \chi (\varPsi _{23}^2 \varPsi _{44}-2 \varPsi _{23} \varPsi _{24} \varPsi _{43}) \\&+ \zeta (\varPsi _{22}^2 \varPsi _{34}+2 \varPsi _{22} \varPsi _{24} \varPsi _{32})\\ \gamma _{(2,0,1,0)}&= 3 \varPsi _{21}^2 \varPsi _{23} \kappa -\zeta \chi (\varPsi _{21}^2 \varPsi _{43}-2 \varPsi _{21} \varPsi _{23} \varPsi _{41}) \\ \gamma _{(2,0,0,1)}&= 3 \varPsi _{21}^2 \varPsi _{24} \kappa - \zeta \chi (\varPsi _{21}^2 \varPsi _{44}-2 \varPsi _{22} \varPsi _{24} \varPsi _{42}) \\&+ \zeta (\varPsi _{22}^2 \varPsi _{34}+2 \varPsi _{22} \varPsi _{24} \varPsi _{32})\\ \gamma _{(0,1,2,0)}&= 3 \varPsi _{22} \varPsi _{23}^2 \kappa - \zeta \chi (\varPsi _{23}^2 \varPsi _{42}-2 \varPsi _{22} \varPsi _{23} \varPsi _{43}) \\&+ \zeta (\varPsi _{23}^2 \varPsi _{32}+2 \varPsi _{22} \varPsi _{23} \varPsi _{33})\\ \gamma _{(1,0,2,0)}&= 3 \varPsi _{21} \varPsi _{23}^2 \kappa -\zeta \chi (\varPsi _{23}^2 \varPsi _{41}-2 \varPsi _{21} \varPsi _{23} \varPsi _{43}) \\&+ \zeta (\varPsi _{23}^2 \varPsi _{31}+2 \varPsi _{21} \varPsi _{23} \varPsi _{33})\\ \gamma _{(0,2,1,0)}&= 3 \varPsi _{22}^2 \varPsi _{23} \kappa - \zeta \chi (\varPsi _{22}^2 \varPsi _{43}-2 \varPsi _{22} \varPsi _{23} \varPsi _{42}) \\&+ \zeta (\varPsi _{22}^2 \varPsi _{33}+2 \varPsi _{22} \varPsi _{23} \varPsi _{32})\\ \gamma _{(2,0,1,0)}&= 3 \varPsi _{21}^2 \varPsi _{23} \kappa - \zeta \chi (\varPsi _{21}^2 \varPsi _{43}-2 \varPsi _{21} \varPsi _{23} \varPsi _{41}) \\&+ \zeta (\varPsi _{21}^2 \varPsi _{33}+2 \varPsi _{21} \varPsi _{23} \varPsi _{31})\\ \gamma _{(1,2,0,0)}&= 3 \varPsi _{21} \varPsi _{22}^2 \kappa -\zeta \chi (\varPsi _{22}^2 \varPsi _{41} -2 \varPsi _{21} \varPsi _{22} \varPsi _{42})\\&+ \zeta (\varPsi _{22}^2 \varPsi _{31}+ 2 \varPsi _{21} \varPsi _{22} \varPsi _{32})\\ \gamma _{(2,1,0,0)}&= 3 \varPsi _{21}^2 \varPsi _{22} \kappa -\zeta \chi (\varPsi _{21}^2 \varPsi _{42} -2 \varPsi _{21} \varPsi _{22} \varPsi _{41})\\&+\zeta (\varPsi _{21}^2 \varPsi _{32}+ 2 \varPsi _{21} \varPsi _{22} \varPsi _{31}) \end{aligned}$$

Remark A.1

It can be noted that $\zeta$ vanishes in the “zero dissipation limit” $\xi _1=0$.