Elastic metamaterial

Remarks

Example: the mass-in-mass cell has an outer mass $m_1$ and an inner mass $m_2$ attached to it by a spring of stiffness $k_2$. Let $\omega_0 = \sqrt{k_2/m_2}$. If the outer mass moves as $x_1 e^{i\omega t}$, the inner mass moves as $x_2 e^{i\omega t}$ with $x_2 = k_2 x_1 / (k_2 - m_2 \omega^2)$. The total force on the cell is $-\omega^2 (m_1 x_1 + m_2 x_2) = -\omega^2 m_{\mathrm{eff}} x_1$ with $$m_{\mathrm{eff}}(\omega) = m_1 + \frac{m_2 \omega_0^2}{\omega_0^2 - \omega^2}.$$ This is negative exactly for $\omega_0 < \omega < \omega_0 \sqrt{1 + m_2/m_1}$, and a chain of such cells has a band gap there.

Non-example: in the long-wavelength limit, a laminate of two ordinary elastic layers has positive effective density and moduli. They are averages of the positive constants of the layers, so clause 2 fails.

The effective constants of clause 1 come from effective medium approximations or from dynamic homogenization. The negative band is caused by the resonance of the inner mass.