# Reduced Linear Constrained Elastic and Viscoelastic Homogeneous Cosserat Media as Acoustic Metamaterials

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## Abstract

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## 1. Introduction

## 2. Model and Basic Laws

#### 2.1. Constitutive Equations for the Linear Elastic Constrained Reduced Cosserat Media

#### 2.2. Isotropic Elastic Model

#### 2.3. Elastic Model with the Simplest Anisotropic Coupling Term (Axial Symmetry with Axis $\mathbf{n}$)

#### 2.3.1. Special Directions of Wave Propagation

#### 2.3.2. Mixed Wave

#### Summary for the Elastic Models

- there is an infinite bandgap above the boundary frequency ${\omega}_{1}$ for the shear waves in both models and above a lower frequency ${\omega}_{1a}$, which depends also on the anisotropy and propagation direction, for the slower mixed wave in the considered anisotropic case.
- compression wave in the isotropic case is classical, non-dispersive
- the upper branch of the mixed wave in the considered anisotropic case is weakly dispersive and has no band gap; its effective moduli, in particular, depend on the direction of the wave propagation and anisotropy parameter

#### 2.4. Isotropic Viscoelastic Model

#### 2.4.1. Plane Shear Wave

#### 2.4.2. Dispersion Relation Properties of the Viscoelastic Model

#### Asymptotics for Various Domains of Frequencies

#### Detailed Analysis of the Real Part of the Wavenumber. Influence of the Dissipation Parameter n on Its Behaviour

- at $0<\omega <{\omega}_{1}$, i.e., in the zone where the shear wave in the elastic medium propagates, as well as at $\omega ={\omega}_{1}$, the value of $\Re k$ decreases when n increases;
- at $\omega >{\omega}_{1}$, i.e., in the zone where the shear wave in the elastic medium does not exist, $\Re k$ for a given $\omega $ first increases as n increases, then reaches its maximal value at $n=\frac{\sqrt{3}({\omega}^{2}-{\omega}_{1}^{2})}{\omega {\omega}_{1}}$ and then decreases as n increases. The corresponding maximal value of ${(\Re k)}^{2}$ is given by$${\left(\frac{{c}_{s}\Re k}{{\omega}_{1}}\right)}^{2}=\frac{{\omega}^{2}}{8({\omega}^{2}-{\omega}_{1}^{2})}.$$

#### Logarithmic Decrement

#### Summary for the Viscoelastic Case

- P-wave is classical due to the isotropy of the model
- Viscoelasticity makes a bandgap for the shear wave to disappear and creates a decreasing part of the dispersion curve
- In the former bandgap (existing for elastic case) viscosity favors the shear wave propagation and below it attenuates the wave
- There exists a boundary wavelength, i.e., a minimal wavelength for propagating waves

## 3. Discussion

## 4. Materials and Methods

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Estimation of the Minimal Wavelength

## Appendix B. Numerical Scripts

`import numpy as np`

`import matplotlib.pyplot as plt`

`import cmath as~cm`

`# assign~values`

`# omega0=1, cs=1, since we have adimensionalised the equations`

`cp = 2.01 # cp^2 = (lambda + 2 mu)/rho, cp --- compression wave velocity`

`# in the isotropic case`

`cn = 2 # cn^2=N/rho. For~isotropic medium it is 0.`

`c = 0.707 # c = cos(k,n) --- cosine of angle between the axis of isotropy`

`# and direction of wave propagation`

`cpnsq = cp**2+2*cn**2*c**2`

`N = 10000 # number of elements to evaluate`

`k_values = np.linspace(0.0,1,N) # wave number`

`real_omegasqp = np.zeros(N) # square of the adimensionalised`

`# frequency, upper part of the dispersion branch.`

`real_omegasqm = np.zeros(N) # square of the adimensionalised`

`# frequency, lower part of the dispersion~branch`

`real_omegap = np.zeros(N) # adimensionalised frequency,`

`# upper part of the dispersion branch. For~isotropic case`

`# it becomes a compressional wave`

`real_omegam = np.zeros(N) # adimensionalised frequency,`

`# lower part of the dispersion branch. For~isotropic case`

`# it is a shear~wave`

`#-----------`

`# MAIN CODE`

`#-----------`

`for o_index in range(N):`

`k = k_values[o_index]`

`# omega1/csalphad=1, we adimensionalised the equations`

`omegasqp = 0.5*k**2*(cpnsq+1/(1+k**2)`

`+cm.sqrt((cpnsq-1/(1+k**2))**2+4*cn**4*c**2*(1-c**2)/(1+k**2)))`

`omegasqm = 0.5*k**2*(cpnsq+1/(1+k**2)`

`-cm.sqrt((cpnsq-1/(1+k**2))**2+4*cn**4*c**2*(1-c**2)/(1+k**2)))`

`real_omegasqp[o_index] = np.real(omegasqp)`

`real_omegasqm[o_index] = np.real(omegasqm)`

`omegap = cm.sqrt(omegasqp)`

`real_omegap[o_index] = np.real(omegap)`

`omegam = cm.sqrt(omegasqm)`

`real_omegam[o_index] = np.real(omegam)`

`#----------------`

`# plot solutions`

`#----------------`

`font = {’size’: 12}`

`plt.rc(’font’, **font)`

`plt.figure(num=1,figsize=(12,6),dpi=100) # define plot size in inches`

`# (width, height) & resolution(DPI)`

`plt.subplot(111)`

`plt.plot(k_values,real_omegap, ’b*’,linewidth=1)`

`plt.plot(k_values,real_omegam,’r*’,linewidth=1)`

`plt.xlabel(’$c_s k/\omega_1$’, size=28)`

`plt.ylabel(’$\omega/\omega_1$’,size=28)`

`plt.grid(True)`

`plt.show()`

`import numpy as np`

`import matplotlib.pyplot as plt`

`import cmath as~cm`

`# viscoelastic isotropic constrained reduced Cosserat medium,`

`# dispersion curves and logarithmic~decrement`

`# assign values`

`n = 0.2`

`# omega1=1, cs=1, since we have adimensionalised the~equations`

`N = 10000 # number of elements to evaluate`

`x_values = np.linspace(0.0,1.5,N) # x = omega / omega_1`

`real_tgphi = np.zeros(N) # tgphi = Re k / Im k`

`real_kdsq = np.zeros(N) # kdsq = (Re k c_s / omega_1)^2`

`real_kd = np.zeros(N) # kd = Re k c_s / omega_1`

`real_kverif = np.zeros(N) # kverif = Im k c_s / omega_1`

`#-----------`

`# MAIN CODE`

`#-----------`

`for o_index in range(N):`

`x = x_values[o_index]`

`kdsq = x**2*((1-x**2)+cm.sqrt((1-x**2)**2+n**2*x**2))/(2*((1-x**2)**2+n**2*x**2))`

`real_kdsq[o_index] = np.real(kdsq)`

`kd = cm.sqrt(kdsq)`

`real_kd[o_index] = np.real(kd)`

`kverif = -n*x**3/(2*kd*(1-x**2)**2+n**2*x**2)`

`real_kverif[o_index] = np.real(kverif)`

`tgphi = kverif/kd`

`real_tgphi[o_index] = np.real(tgphi)`

`#----------------`

`# plot solutions`

`#----------------`

`font = {’size’: 12}`

`plt.rc(’font’, **font)`

`plt.figure(num=1,figsize=(12,6),dpi=100) # define plot size in inches`

`(width, height) & resolution(DPI)`

`plt.subplot(111)`

`plt.plot(real_kd, x_values, ’b*’,linewidth=1)`

`plt.plot(real_kverif,x_values,’r--’,linewidth=3)`

`plt.plot(-real_tgphi,x_values,’c-.’,linewidth=4)`

`plt.xlabel(’Solid: Re $c_s k/\omega_1$, dashed: Im $c_s k/\omega_1$,`

`dashed-dotted: |Im $k$ / Re $k$|, n=’+str(n), size=28)`

`#plt.xlabel(’Re $c_s k/\omega_1,\quad$’+chr(957)+’$\omega_1=$’+str(n), size=28)`

`#plt.xlabel(’Im $c_s k/\omega_0$’, color=’red’, size=16)`

`plt.ylabel(’$\omega/\omega_1$’,size=28)`

`#plt.ylabel(Blue: ’Re $k$, red: Im $k$’,size=16)`

`plt.grid(True)`

`plt.show()`

`plt.xlabel(’Solid: Re $c_s k/\omega_1$, dashed: Im $c_s k/\omega_1$,`

`dashed-dotted: |Im $k$ / Re $k$|, n=’+str(n), size=28)`

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**Figure 1.**Dispersion graph for the plane shear–rotational wave in the 3D infinite elastic isotropic linear reduced constrained Cosserat medium. Parameters: ${c}_{s}=1$. Infinite band gap above ${\omega}_{1}$.

**Figure 2.**Dispersion curves for the constrained anisotropic elastic linear reduced Cosserat medium, ${c}_{P}=2.01,{c}_{N}=2,\widehat{\mathbf{k}}\xb7\mathbf{n}=0.707$. Both curves are dispersive but for the upper branch we hardly notice this.

**Figure 3.**Dispersion curve for the linear constrained reduced viscoelastic isotropic Cosserat medium, $\frac{\omega}{{\omega}_{1}}$ vs. $\Re \frac{{c}_{s}k}{{\omega}_{1}},\Im \frac{{c}_{s}k}{{\omega}_{1}}$, $n=0.05$.

**Figure 4.**Dispersion curve for the linear constrained reduced viscoelastic isotropic Cosserat medium, $\frac{\omega}{{\omega}_{1}}$ vs. $\Re \frac{{c}_{s}k}{{\omega}_{1}}$, $n=0.05$.

**Figure 5.**Dispersion curve for the linear constrained reduced viscoelastic isotropic Cosserat medium, $\frac{\omega}{{\omega}_{1}}$ vs. $\Re \frac{{c}_{s}k}{{\omega}_{1}}$, $n=0.5$.

**Figure 6.**Dispersion curve for the linear constrained reduced viscoelastic isotropic Cosserat medium, $\frac{\omega}{{\omega}_{1}}$ vs. $\Re \frac{{c}_{s}k}{{\omega}_{1}}$, $n=2$.

**Figure 7.**Dispersion curve for the linear constrained reduced viscoelastic isotropic Cosserat medium, $\frac{\omega}{{\omega}_{1}}$ vs. $\Re \frac{{c}_{s}k}{{\omega}_{1}}$, $n=5$.

**Figure 8.**Dispersion curve for the shear plane wave in the linear constrained reduced viscoelastic isotropic Cosserat medium, $\frac{\omega}{{\omega}_{1}}$ vs: $\Re \frac{{c}_{s}k}{{\omega}_{1}}$ (solid), $\Im \frac{{c}_{s}k}{{\omega}_{1}}$ (dashed), logarithmic decrement divided by $2\pi $ (dashed-dotted), $n=0.2$.

Medium | Wave Properties | Classification |
---|---|---|

Isotropic elastic | Classical compression wave. Highly dispersive shear–rotational wave, which does not propagate above boundary frequency ${\omega}_{1}$ (large wavenumber limit) | single negative acoustic metamaterial at $\omega >{\omega}_{1}$ |

Elastic with a simplest axisymmetric coupling $N\nabla \xb7\mathbf{u}(\mathbf{n}\xb7\nabla \mathbf{u}\xb7\mathbf{n})$ between shear and volumetric strains | Shear wave (same as in isotropic case), non-dispersive compression wave (for $\mathbf{k}\perp \mathbf{n}$ with classical velocity or $\mathbf{k}||\mathbf{n}$ with larger velocity), and two mixed waves coexist. One mixed wave is weakly, the other is strongly dispersive with a boundary frequency ${\omega}_{1a}<{\omega}_{1}$ depending on the propagation direction | single negative acoustic metamaterial with respect to the shear wave at $\omega >{\omega}_{1}$ and with respect to the slower mixed wave at $\omega >{\omega}_{1a}$ |

Viscoelastic isotropic | Classical compression wave. Highly dispersive shear–rotational wave. Its dispersion branch has a decreasing part and large velocity at large $\omega $; there is a minimal wavelength, $\Re k(\omega )<{k}_{max}(n)$ has at least one maximum; at small $\omega $ the wave tends to classical; band gap disappeared, in its domain dissipation enhances wave propagation, at $\omega <{\omega}_{1}$ attenuates the wave, at ${\omega}_{1}$ does not influence the attenuation Logarithmic decrement is proportional to $\omega $ at low and high frequencies (proved analytically and checked numerically). $\Re k(\omega )$ has only one maximum, the group velocity there is infinite, this point separates zones of normal and anomalous dispersion.$\Im k$ has one maximum (at small n) or does not have, it depends on n in a complex way (checked numerically). | Double negative acoustic metamaterial with respect to the shear wave above a certain frequency |

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## Share and Cite

**MDPI and ACS Style**

Grekova, E.F.; Porubov, A.V.; dell’Isola, F.
Reduced Linear Constrained Elastic and Viscoelastic Homogeneous Cosserat Media as Acoustic Metamaterials. *Symmetry* **2020**, *12*, 521.
https://doi.org/10.3390/sym12040521

**AMA Style**

Grekova EF, Porubov AV, dell’Isola F.
Reduced Linear Constrained Elastic and Viscoelastic Homogeneous Cosserat Media as Acoustic Metamaterials. *Symmetry*. 2020; 12(4):521.
https://doi.org/10.3390/sym12040521

**Chicago/Turabian Style**

Grekova, Elena F., Alexey V. Porubov, and Francesco dell’Isola.
2020. "Reduced Linear Constrained Elastic and Viscoelastic Homogeneous Cosserat Media as Acoustic Metamaterials" *Symmetry* 12, no. 4: 521.
https://doi.org/10.3390/sym12040521