Theoretical limits for negative elastic moduli in sub-acoustic lattice materials (original) (raw)

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 behaviour 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 sub-wavelength 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. Introduction.-The global mechanical properties can be engineered in lattice materials by intelligently identifying the material microstructures as the properties in these materials are often defined by their structural configuration along with the intrinsic material properties of the constituent members. This novel class of materials with tailorable application-specific mechanical properties (like equivalent elastic moduli, buckling, vibration and wave propagation characteristics with modulation features) have tremendous potential applications for future aerospace, civil, mechanical, electronics and medical applications across the length-scales. Naturally occurring materials cannot exhibit unprecedented and fascinating properties such as extremely lightweight, negative elastic moduli, negative mass density, pentamode material characteristics (meta-fluid), which can be achieved by an intelligent microstructural design [1, 2]. For example, the conventional positive value of Poisson's ratio in a hexagonal lattice metamaterial can be converted to a negative value [3] by making the cell angle θ in figure 1(b) negative. Other unusual and exciting properties can be realized in metamaterials under dynamic condition, such as negative bulk modulus induced by monopolar resonance [4], negative mass density induced by dipolar resonance [5], and negative shear modulus induced by quadrupolar resonance [6]. Elastic cloaks [7] and various other unprecedented dynamic behaviour of such materials have been widely reported in literature [8-14].