A nonlinear negative stiffness metamaterial unit cell and small-on-large multiscale material model

Abstract

A persistent challenge in the design of composite materials is the ability to fabricate materials that simultaneously display high stiffness and high loss factors for the creation of structural elements capable of passively suppressing vibro-acoustic energy. Relevant recent research has shown that it is possible to produce composite materials whose macroscopic mechanical stiffness and loss properties surpass those of conventional composites through the addition of trace amounts of materials displaying negative stiffness (NS) induced by phase transformation [R. S. Lakes et al., Nature 410, 565-567 (2001)]. The present work investigates the ability to elicit NS behavior without employing physical phenomena such as inherent nonlinear material behavior (e.g., phase change or plastic deformation) or dynamic effects, but rather the controlled buckling of small-scale structural elements, metamaterials, embedded in a continuous viscoelastic matrix. To illustrate the effect of these buckled elements, a nonlinear hierarchical multiscale material model is derived, which estimates the macroscopic stiffness and loss of a composite material containing pre-strained microscale structured inclusions. The multiscale model consists of two scale transition models, the first being an energy-based nonlinear finite element (FE) method to determine the tangent modulus of the metamaterial unit cell, and the other a classical analytical micromechanical model to determine the effective stiffness and loss tensors of a heterogeneous material for small perturbations from the local strain state of the unit cells. The FE method enables the estimation of an effective nonlinear anisotropic stiffness tensor of a buckled microstructure that produces NS and is sufficiently general to consider geometries different from those given in this work.