Theoretical limits for negative elastic moduli in subacoustic lattice materials

An insightful mechanics-based bottom-up framework is developed for probing the frequency dependence of lattice material microstructures. Under a vibrating condition, effective elastic moduli of such microstructured materials can become negative for certain frequency values, leading to an unusual mechanical behavior with a multitude of potential applications. We have derived the fundamental theoretical limits for the minimum frequency, beyond which the negative effective moduli of the materials could be obtained. An efficient dynamic stiffness matrix based approach is developed to obtain the closed-form limits, which can exactly capture the subwavelength scale dynamics. The limits turn out to be a fundamental property of the lattice materials and depend on certain material and geometric parameters of the lattice in a unique manner. An explicit characterization of the theoretical limits of negative elastic moduli along with adequate physical insights would accelerate the process of its potential exploitation in various engineered materials and structural systems under dynamic regime across the length scales.

Figure 1

Bottom-up approach (involving a hierarchy of analysis with beam element, unit cell, and lattice structure) for analyzing the frequency-dependent elastic moduli of lattice materials. (a) Typical representation of a hexagonal cellular structure in a dynamic environment (such as the honeycomb as part of a host structure experiencing wave propagation, vibrating structural component, etc.). The curved arrows are symbolically used to indicate propagation of wave. (b) One hexagonal unit cell under dynamic environment. (c) A dynamic beam element for the damped bending vibration with two nodes and four degrees of freedom.

Figure 2

Real and imaginary parts and the amplitude of the normalized value of $E_1$ and $E_2$ as a function of frequency. Here the first frequency value when the Young's moduli become negative is marked by “+”.

Figure 3

Real and imaginary parts and the amplitude of the normalized value of $G_{12}$ as a function of frequency for two different values of $h/l$. Bounds of ${\omega }_{G_{12}}^{*}$, the frequency at which the $G_{12}$ becomes negative, are calculated from Eq. (8). Here the upper and lower bounds of frequency where $G_{12}$ becomes negative for the first time are shown by “$\times$” and “+”, respectively.