Effective Description of Anisotropic Wave Dispersion in Mechanical Band-Gap Metamaterials via the Relaxed Micromorphic Model

Abstract

In this paper the relaxed micromorphic material model for anisotropic elasticity is used to describe the dynamical behavior of a band-gap metamaterial with tetragonal symmetry. Unlike other continuum models (Cauchy, Cosserat, second gradient, classical Mindlin–Eringen micromorphic etc.), the relaxed micromorphic model is endowed to capture the main microscopic and macroscopic characteristics of the targeted metamaterial, namely, stiffness, anisotropy, dispersion and band-gaps.

The simple structure of our material model, which simultaneously lives on a micro-, a meso- and a macroscopic scale, requires only the identification of a limited number of frequency-independent and thus truly constitutive parameters, valid for both static and wave-propagation analyses in the plane. The static macro- and micro-parameters are identified by numerical homogenization in static tests on the unit-cell level in Neff et al. (J. Elast., https://doi.org/10.1007/s10659-019-09752-w, 2019, in this volume).

The remaining inertia parameters for dynamical analyses are calibrated on the dispersion curves of the same metamaterial as obtained by a classical Bloch–Floquet analysis for two wave directions.

We demonstrate via polar plots that the obtained material parameters describe very well the response of the structural material for all wave directions in the plane, thus covering the complete panorama of anisotropy of the targeted metamaterial.

Appendix: Equivalent Dynamical Determination of the Macroscopic Stiffness \(\mathbb{C}_{\textrm{macro}}\)

In this appendix we show how the macroscopic parameters previously obtained by simple static arguments can be equivalently computed using the slopes of the acoustic dispersion curves close to the origin. This equivalent method is very useful to make a strong connection between the relaxed micromorphic model and classical elasticity, but it does not add any extra feature to the fitting procedure presented above.

1.1 Tangents in Zero to the Acoustic Branches

Let us consider the “macroscopic” Cauchy partial differential system of equations

$$ \rho u_{,\mathit{tt}}-\textrm{Div} (\mathbb{C}_{\textrm{macro}}\operatorname{sym}\nabla u)=0 $$

(19)

which is the limiting case of the relaxed micromorphic model when $L_{c}$ tends to zero. In this classical case, it is possible to obtain an analytical expression for the dispersion curves. In order to achieve this goal, we insert a generic plane wave function $\widetilde{u} e^{\mathfrak{i}( \langle \boldsymbol{k},x \rangle-\omega t )}$ in (19) obtaining

$$\begin{aligned} &\textrm{Div} \bigl[\mathbb{C}_{\textrm{macro}}\operatorname{sym}\nabla \bigl(\widetilde{u} e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle-\omega t )} \bigr) \bigr] \\ & \quad = \biggl[(\mathbb{C}_{\textrm{macro}})_{ijmn} \frac{1}{2} ( \widetilde{u}_{m}k_{n}+ \widetilde{u}_{n}k_{m} ) \mathfrak{i}e^{\mathfrak{i}( \langle\boldsymbol {k},x \rangle-\omega t )} \biggr]_{,j} \\ & \quad =-\frac{1}{2} \bigl[(\mathbb{C}_{\textrm{macro}})_{ijmn} ( \widetilde{u}_{m}k_{n}+\widetilde{u}_{n}k_{m} ) k_{j} e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle-\omega t )} \bigr] \\ & \quad =-\frac{1}{2} \bigl[ (\mathbb{C}_{\textrm{macro}})_{ijmn}k_{j} \widetilde{u}_{m}k_{n}+ (\mathbb{C}_{\textrm{macro}})_{ijmn}k_{j} \widetilde{u}_{n}k_{m} \bigr]e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle-\omega t )} \\ & \quad =-\frac{1}{2} \bigl[ (\mathbb{C}_{\textrm{macro}})_{ijmn}k_{j} \widetilde{u}_{m}k_{n}+ (\mathbb{C}_{\textrm{macro}})_{ijnm}k_{j} \widetilde{u}_{n}k_{m} \bigr]e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle-\omega t )} \\ & \quad =- (\mathbb{C}_{\textrm{macro}})_{ijmn}k_{j}\widetilde{u}_{m}k_{n} e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle-\omega t )}, \end{aligned}$$

(20)

for the static term and

$$\rho u_{,\mathit{tt}}=-\rho \omega^{2} \widetilde{u} e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle-\omega t )}=-\rho \omega^{2} \delta_{\mathit{im}} \widetilde{u}_{m} e^{\mathfrak{i}( \langle \boldsymbol{k},x \rangle-\omega t )} $$

for the kinetic term. Thus, finally combining the two expressions we have that $\widetilde{u} e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle-\omega t )}$ satisfies the system (19) if and only if

$$-\rho \omega^{2} \delta_{\mathit{im}} \widetilde{u}_{m} e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle-\omega t )}+ (\mathbb{C}_{\textrm{macro}})_{\mathit{ijmn}}k_{j} \widetilde{u}_{m}k_{n} e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle-\omega t )}=0 $$

and since the function $e^{\mathfrak{i}( \langle\boldsymbol{k},x \rangle -\omega t )}$ is always non-zero this gives

$$ \bigl(\rho \omega^{2}\delta_{\mathit{im}}- (\mathbb{C}_{\textrm{macro}}){}_{\mathit{ijmn}} k_{j} k_{n} \bigr) \widetilde{u}_{m}=0 $$

(21)

and writing the wave vector as $\boldsymbol{k}=k\widehat{\boldsymbol {k}}$ with $\widehat{\boldsymbol{k}}= (\widehat{k}_{1},\widehat {k}_{2},\widehat{k}_{3} )$ unitary and $k= \Vert \boldsymbol{k} \Vert$, we arrive to

$$ \bigl(\rho\omega^{2}\delta _{\mathit{im}}-k^{2} (\mathbb{C}_{\textrm{macro}}){}_{\mathit{ijmn}} \widehat{k}_{j} \widehat{k}_{n} \bigr) \widetilde{u}_{m}=0. $$

(22)

The equation (22) is an eigenvalues problem for the linear application $k^{2} (\mathbb{C}_{\textrm{macro}}){}_{\mathit {ijmn}}\widehat{k}_{j}\widehat{k}_{n}$. The stated problem admits non-trivial solutions if and only if the determinant of $\rho\omega^{2}\delta_{\mathit{im}}-k^{2} (\mathbb{C}_{\textrm{macro}}){}_{\mathit{ijmn}}\widehat{k}_{j}\widehat{k}_{n}$ is zero. In this way, we are interested in looking for couples $(k,\omega )$ such that

$$ \det \bigl(\rho\omega^{2}\delta_{\mathit{im}}-k^{2} (\mathbb{C}_{\textrm{macro}}){}_{\mathit{ijmn}}\widehat{k}_{j}\widehat{k}_{n} \bigr)=0. $$

(23)

Moreover, we are interested in studying this problem as a function of the direction of propagation $\widehat{\boldsymbol{k}}$ of the wave. In order to do this, it is convenient to introduce spherical coordinates for the wave vector $\widehat{\boldsymbol{k}}\in\mathbb{S}^{2}$ (the unit sphere in $\mathbb{R}^{3}$):

$$ k_{1}=\sin\varphi\cos\vartheta,\qquad k_{2}=\sin \varphi\sin\vartheta,\qquad k_{3}=\cos\varphi, $$

(24)

where $\vartheta\in [0,2\pi )$ is the polar angle and $\varphi\in [0,\pi ]$ is the azimuthal angle. For the problem in the $(x_{1},x_{2},0 )$ plane, the angle $\varphi$ is $\pi/2$, so

$$ k_{1}=\cos\vartheta, \qquad k_{2}=\sin\vartheta, \qquad k_{3}=0, $$

(25)

and

$$\widehat{\boldsymbol{k}}\otimes\widehat{\boldsymbol{k}}= \begin{pmatrix}\cos^{2}\vartheta& \cos\vartheta\sin\vartheta& 0\\ \cos\vartheta\sin\vartheta& \sin^{2}\vartheta& 0\\ 0 & 0 & 0 \end{pmatrix} . $$

Let us now consider the Voigt representation of the tensor $\mathbb {C}_{\textrm{macro}}$ in the case of the tetragonal symmetry

$$\begin{aligned} &\widetilde{\mathbb{C}}_{\textrm{macro}} \\ & \quad = \small{ \left ( \textstyle\begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c} 2\mu_{\textrm{macro}}+\lambda_{\textrm{macro}} & \lambda_{\textrm{macro}} & \lambda_{\textrm{macro}}^{*} & 0 & 0 & 0\\ \lambda_{\textrm{macro}} & 2\mu_{\textrm{macro}}+\lambda_{\textrm {macro}} & \lambda_{\textrm{macro}}^{*} & 0 & 0 & 0\\ \lambda_{\textrm{macro}}^{*} & \lambda_{\textrm{macro}}^{*} & (\widetilde{\mathbb{C}}_{\textrm{macro}})_{33} & 0 & 0 & 0\\ 0 & 0 & 0 & (\widetilde{\mathbb{C}}_{\textrm{macro}})_{44} & 0 & 0\\ 0 & 0 & 0 & 0 & (\widetilde{\mathbb{C}}_{\textrm{macro}})_{44} & 0\\ 0 & 0 & 0 & 0 & 0 & \mu_{\textrm{macro}}^{*} \end{array}\displaystyle \right ) } . \end{aligned}$$

A direct calculation gives

$$\begin{aligned} &(\mathbb{C}_{\textrm{macro}}){}_{\mathit{ijmn}}\widehat{k}_{j}\widehat{k}_{n} \\ & = \left ( \textstyle\begin{array}{c@{\ }c@{\ }c} \mu_{\textrm{macro}}^{\ast}\,\sin^{2}\vartheta+\cos^{2}\vartheta (\lambda_{\textrm{macro}}+2\mu_{\textrm{macro}} ) & \cos\vartheta \sin\vartheta (\mu_{\textrm{macro}}^{\ast}+\lambda_{\textrm {macro}} ) & 0\\ \cos\vartheta\sin\vartheta (\mu_{\textrm{macro}}^{\ast}+\lambda _{\textrm{macro}} ) & \cos^{2}\vartheta\mu_{\textrm{macro}}^{\ast }+\sin^{2}\vartheta (\lambda_{\textrm{macro}}+2\mu_{\textrm{macro}} ) & 0\\ 0 & 0 & (\widetilde{\mathbb{C}}_{\textrm{macro}} )_{44} \end{array}\displaystyle \right ), \end{aligned}$$

thus, for $\rho\omega^{2}\delta_{\mathit{im}}-k^{2} (\mathbb{C}_{\textrm{macro}}){}_{\mathit{ijmn}}\widehat{k}_{j}\widehat{k}_{n}$, we find

$$\begin{aligned} &\left ( \textstyle\begin{array}{c} \rho\omega^{2}-k^{2} (\mu_{\textrm{macro}}^{\ast}\sin^{2}\vartheta +\cos^{2}\vartheta (\lambda_{\textrm{macro}}+2\mu_{\textrm{macro}} ) ) \\ -k^{2}\cos\vartheta\sin\vartheta (\mu_{\textrm{macro}}^{\ast }+\lambda_{\textrm{macro}} ) \\ 0 \end{array}\displaystyle \right. \\ & \quad \left . \textstyle\begin{array}{c@{\quad}c} -k^{2}\cos\vartheta\sin\vartheta (\mu_{\textrm{macro}}^{\ast }+\lambda_{\textrm{macro}} ) & 0 \\ \rho\omega^{2}-k^{2} (\mu_{\textrm{macro}}^{\ast }\cos^{2}\vartheta+\sin^{2}\vartheta (\lambda_{\textrm{macro}}+2\mu_{\textrm{macro}} )) &0 \\ 0 &\rho\omega^{2}- (\widetilde{\mathbb{C}}_{\textrm{macro}} )_{44}k^{2} \end{array}\displaystyle \right ). \end{aligned}$$

In this way we can compute

$$\begin{aligned} &\det \bigl(\rho\omega^{2}\delta_{im}-k^{2} (\mathbb{C}_{\textrm{macro}}){}_{ijmn}\widehat{k}_{j}\widehat{k}_{n} \bigr) \\ & \quad = \bigl(\rho\omega^{2}- (\widetilde{\mathbb{C}}_{\textrm{macro}} )_{44}k^{2} \bigr) \bigl[ \bigl(\rho\omega ^{2}-k^{2} \bigl(\cos^{2}\vartheta(\lambda_{\textrm{macro}}+2 \mu_{\textrm{macro}})+\mu_{\textrm{macro}}^{\ast}\sin^{2}\vartheta \bigr) \bigr) \\ & \qquad \times \bigl(\rho\omega^{2}-k^{2} \bigl(\sin ^{2}\vartheta(\lambda_{\textrm{macro}}+2\mu_{\textrm{macro}})+\mu_{\textrm{macro}}^{\ast}\cos ^{2} \vartheta \bigr) \bigr) \\ &\qquad {}-k^{4}\sin^{2} \vartheta\cos^{2} \vartheta \bigl(\lambda_{\textrm{macro}}+\mu_{\textrm{macro}}^{\ast} \bigr){}^{2} \bigr]. \end{aligned}$$

(26)

The dispersion curves for the classical limit Cauchy model are obtained solving the equation (23), or equivalently (26), with respect to $\omega^{2}$. We call $\{ \pm\omega_{\textrm {macro},i} (k,\vartheta ) \} _{i=1}^{3}$ the dispersion curves for the Cauchy continuum obtained by solving (23) for the special tetragonal case given in (26). A direct calculation shows that the positive solutions $\{ \omega _{\textrm{macro},i} (k,\vartheta ) \} _{i=1}^{3}$ of (23) for the tetragonal case are three straight lines in the $(\omega,k )$ plane with slopes (here, and only here, we use the abbreviations $\mu_{M}=\mu_{\textrm{macro}}$, $\lambda_{M}=\lambda_{\textrm{macro}}$, $\mu_{M}^{*}=\mu_{\textrm{macro}}^{*}$):

$$\begin{aligned} a_{\mathrm{LA}} & =\sqrt{\frac{\lambda_{M}+\mu_{M}^{\ast}+2\mu_{M}+\sqrt{2\cos(4\vartheta)\left(\lambda_{M}+\mu_{M}\right)\left(\mu_{M}-\mu_{M}^{\ast}\right)+2\lambda_{M}\mu_{M}+\lambda_{M}^{2}+\mu_{M}^{\ast2}-2\mu_{M}\mu_{M}^{\ast}+2\mu_{M}^{2}}}{2\rho}}, \end{aligned}$$

(27)

$$\begin{aligned} a_{\mathrm{TA}} & =\sqrt{\frac{\lambda_{M}+\mu_{M}^{\ast}+2\mu_{M}-\sqrt{2\cos(4\vartheta)\left(\lambda_{M}+\mu_{M}\right)\left(\mu_{M}-\mu_{M}^{\ast}\right)+2\lambda_{M}\mu_{M}+\lambda_{M}^{2}+\mu_{M}^{\ast2}-2\mu_{M}\mu_{M}^{\ast}+2\mu_{M}^{2}}}{2\rho}}, \\ a_{\mathrm{TA}3} & =\sqrt{\frac{\left(\widetilde{\mathbb{C}}_{M}\right)_{44}}{\rho}}. \end{aligned}$$

(28)

Note the complete absence of the Cosserat couple modulus $\mu_{c}$ in the latter formulas.

Such dispersion curves are called (in-plane) longitudinal-acoustic (LA), (in-plane) transverse-acoustic (TA) and (out-of-plane) transverse-acoustic (TA3) (see Fig. 9). Since in this paper we are interested only in vibrations in the $(x_{1},x_{2},0 )$ plane, the third acoustic line with slope $a_{\mathrm{TA}3}$ will not be considered for the fitting procedure since it corresponds to out-of-plane vibrations.

Fig. 9

Dispersion branches for the limiting tetragonal Cauchy continuum with $\widehat{\boldsymbol{k}}= (1,0,0 )$(a) and $\widehat {\boldsymbol{k}}= (\sqrt{2}/2,\sqrt{2}/2,0 )$(b)

One remarkable property of the relaxed micromorphic model is that the slopes of its acoustic curves close to the origin, are exactly given by the slopes of the acoustic lines (27) of the equivalent Cauchy continuum. More precisely, the slopes at zero of the acoustic branches of the dispersion curves as obtained via the relaxed micromorphic model can be computed by means of equations (27) when using the identities (17).

1.2 Dynamical Calculation of the Macroscopic Stiffness $\mathbb{C}_{\textrm{macro}}$

Based on the results of the previous Paragraph, we can compute the numerical values of the macroscopic parameters $\mu_{\textrm{macro}},\lambda_{\textrm{macro}}$ and $\mu_{\textrm{macro}}^{*}$ which represent the measure of the macroscopic stiffness of the considered tetragonal metamaterial. To this aim, considering the two directions of propagation $\vartheta_{0}=0$ and $\vartheta _{1}=\pi/4$, and accounting for the following assumptions on the involved constitutive parameters

$$\lambda_{\textrm{macro}}>0,\quad \mu_{\textrm{macro}}>0,\quad \mu_{\textrm{macro}}^{*}>0,\quad \lambda_{\textrm{macro}}+2\mu_{\textrm{macro}}- \mu_{\textrm{macro}}^{*}>0, $$

we set up the following system of algebraic equations:

$$\begin{aligned} \begin{aligned}[c] a_{\textrm{LA}} \bigl(\vartheta_{0},\lambda_{\textrm{macro}}, \mu_{\textrm{macro}},\mu_{\textrm{macro}}^{*} \bigr)=\sqrt{\frac{2\mu_{\textrm{macro}}+\lambda_{\textrm{macro}}}{\rho}} & = \overline{a}_{\textrm{LA}} (\vartheta_{0} ), \\ a_{\textrm{LA}} \bigl(\vartheta_{1},\lambda_{\textrm{macro}},\mu_{\textrm{macro}},\mu_{\textrm{macro}}^{*} \bigr)=\sqrt{\frac{\mu_{\textrm{macro}}+\mu_{\textrm{macro}}^{*}+\lambda_{\textrm{macro}}}{\rho}} & =\overline {a}_{\textrm{LA}} (\vartheta_{1} ), \\ a_{\textrm{TA}} \bigl(\vartheta_{0},\lambda_{\textrm{macro}},\mu_{\textrm{macro}},\mu_{\textrm{macro}}^{*} \bigr)=\sqrt{\frac{\mu_{\textrm{macro}}^{*}}{\rho}} & =\overline {a}_{\textrm{TA}} (\vartheta_{0} ), \\ a_{\textrm{TA}} \bigl(\vartheta_{1},\lambda_{\textrm{macro}},\mu_{\textrm{macro}},\mu_{\textrm{macro}}^{*} \bigr)=\sqrt{\frac{\mu_{\textrm{macro}}}{\rho}} & =\overline {a}_{\textrm{TA}} (\vartheta_{1} ). \end{aligned} \end{aligned}$$

(29)

This system of algebraic equations counts 4 equations and 3 unknowns. We use the first 3 equations to calculate the unknowns $\lambda_{\textrm{macro}},\mu_{\textrm{macro}},\mu_{\textrm{macro}}^{*}$ and then plug the found values in the fourth equation. If our hypothesis according to which the metamaterial we are considering has a tetragonal symmetry is correct, the fourth equation has to be automatically satisfied. This is indeed the case.

The numerical values of the macroscopic parameters which are found with the described procedure are given in Table 7.

Table 7 Left: numerical values of the macroscopic parameters of the relaxed micromorphic model as obtained via the dynamical fitting and Bloch–Floquet analysis. Right: for comparison the values obtained by periodic homogenization