Modeling and Homogenization of Auxetic Structures Using Higher Order Micropolar Continuum Theory

Abstract Repeating 2D auxetic patterns can be combined to form a continuous flexible sheet capable of high curvature and unique shaping capabilities. Typical auxetic designs consist of a connecting series of linear or curved struts, such as the commonly studied re-entrant auxetic honeycomb. These designs are tunable by controlling their geometric and material parameters. These parameters alter the macro stiffness behavior of the structure, which is important for meeting stretching and bending criterium determined from its intended application. If a high curvature is required, the auxetic pattern may need ample spacing between struts to delay self-collision due to bending. If significant elongation is the goal, the pattern may require many zig-zag struts per unit cell. The challenge in predicting deformation behavior of such structures lies in the difficulty in establishing design-property-deformation relationships. Prior studies have utilized explicit finite element representation of various auxetic patterned sheets to establish such relationships. Such modeling methods are not scalable to larger structures, as the computational costs are prohibitive. This study focuses on developing a homogenization method for auxetic patterns scalable to large structures. Higher order micropolar continuum theory is considered which takes into account the higher order deformation modes of auxetic structures. The method outlined allows for obtaining the micropolar material properties (constitutive relation) associated with axial, shear, bending, and torsional stiffness of various auxetic pattens. This study sets the conditions for modeling such structures in a computationally efficient manner as a micropolar continuum.