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Additively manufactured auxetics (structures exhibiting a negative Poisson’s ratio) offer a unique combination of enhanced mechanical strength and energy absorption. These properties can be further improved through strategic material placement and architectural design. This study investigates the feasibility of fabricating bi-material rotating-square auxetic structures composed of flexible and rigid constituents in their squares and hinges. Rotating-square auxetic structures are manufactured via material extrusion using rigid polylactic acid (PLA) and flexible thermoplastic polyurethane (TPU) to explore the effects of material distribution on mechanical performance and failure characteristics at the macro (i.e., component) and meso (i.e., cell) scales. Baseline tests are conducted to quantify single- and bi-material interfacial strength and failure modes under normal, shear, and combined loading conditions. Upon validation of interface integrity, single- and bi-material auxetic structures are fabricated and tested in uniaxial compression. Relative to the TPU single-material structure, the PLA square-TPU hinge structure provides a 33% increase in structural stiffness, increases energy absorption, delays the global densification strain by 10%, yields a structural Poisson’s ratio at least 0.3 lower than its single-material counterpart through global axial strains of 20%, and demonstrates partial shape recovery. Multiscale experimental analyses supplemented by a kinematic model reveal the rotation-dependent stiffening mechanisms of these structures, highlighting the benefits of flexible hinge materials. Bi-material structures with flexible hinges are shown to have bilinear trends in structural stiffness and energy absorption, not intrinsic to their single-material counterparts. These findings highlight the potential of bi-material design strategies in advancing the functionality and tunability of auxetic structures for the next generation of mechanical metamaterials. 

Auxetic (negative Poisson’s ratio) structures made from rotating squares have attracted considerable attention due to their tunable shape control, strength, and strain energy absorption capacity. The present study aims to explore the interrelations between mesoscale kinematics and the macroscopic mechanical behavior of additively manufactured rotating-square auxetics under compressive loads. Specifically, correlations between the rotational degree of freedom of the squares, mechanical deformation of the cell hinges, and the macroscopic nonlinear mechanical and Poisson’s behaviors are investigated using experimental measurements supplemented by mathematical models. Structures with variable cell hinge thicknesses are fabricated by stereolithography additive manufacturing technique and then subjected to compressive loads applied at quasi-static and dynamic conditions with several orders of magnitude difference in strain rate. Multiscale mechanical deformation of the structure in each case is analyzed using digital image correlation (DIC). Experimental characterizations indicate strongly nonlinear and rate-sensitive auxetic behaviors in the examined structures. The role of cell hinge thickness is discussed in terms of the mechanical constraint that these components impose on the rotational degree of freedom of the solid squares in the structure, concurrently causing a nonlinear strain hardening behavior. 

This study provides an in-depth analysis of the mechanical behavior of rotating-square auxetic structures under various strain rates. The structures are fabricated using stereolithography additive manufacturing with a flexible resin. Mechanical tests performed on structures include quasi-static, intermediate, and high strain rate compression tests, supplemented by high-speed optical imaging and two-dimensional digital image correlation analyses. In quasi-static conditions (5 × 10–3 s-1), multiscale measurements reveal the correlation between local and global strains. It is shown that cell hinges play a significant role in structural deformation and load-bearing capacity. In drop tower impact conditions (intermediate strain rate of ca. 200 s-1), the auxetic structures display significant strain rate hardening compared to loading at quasi-static rates. The thin-hinge structures maintain a Poisson's ratio of approximately -0.8, showing higher auxeticity than slow-rate compression tests. High strain rate conditions (ca. 2000s-1) activate additional deformation mechanisms, including a delayed state of equilibrium exemplified by a heterogeneous distribution of lateral strains, possibly due to stress wave interactions and inertial stresses. The study further reveals nonlinear correlations between Poisson's ratio, strain, and strain rate, indicating reduced auxeticity at higher strain rates. These observations are discussed in terms of complex wave interactions and the strain rate hardening characteristics of the base polymer. 

Abstract Zero Poisson’s ratio structures are a new class of mechanical metamaterials wherein the absence of lateral deformations allows the structure to adapt and conform their geometries to desired shapes with minimal interventions. These structures have gained attention in large deformation applications where shape control is a key performance attribute, with examples including but not limited to shape morphing, soft robotics, and flexible electronics. The present study introduces an experimentally driven approach that leads to the design and development of (near) zero Poisson’s ratio structures with considerable load-bearing capacities through concurrent density and architecture gradations in hybrid honeycombs created from hexagonal and re-entrant cells. The strain-dependent Poisson’s ratios in hexagonal and re-entrant honeycombs with various cell wall thicknesses have been characterized experimentally. A mathematical approach is then proposed and utilized to create hybrid structures wherein the spatial distribution of different cell shapes and densities leads to the development of honeycombs with minimal lateral deformations under compressive strains as high as 0.7. Although not considered design criteria, the load-bearing and energy absorption capacities of the hybrid structures are shown to be comparable with those of uniform cell counterparts. Finally, the new hybrid structures indicate lesser degrees of instability (in the form of cell buckling and collapse) due to the self-constraining effects imposed internally by the adjacent cell rows in the structures. 

Midsoles are important components in footwear as they provide shock absorption and stability, thereby improving comfort and effectively preventing certain foot injuries. A strategically engineered midsole designed to mitigate plantar pressure can enhance athletic performance and comfort levels. Despite the importance of midsole design, the potential of using in-plane density gradation (deliberate variation of material density across the horizontal plane) in midsoles has been rarely explored. The present work investigated the effectiveness of in-plane density gradation in shoe midsoles using novel polyurea foams as the material candidate. Different polyurea foam densities, ranging from 95 to 350 kg/m²were examined and tested to construct density-dependent correlative mathematical relations required for optimizing the midsole design for enhanced cushioning and reduced weight. This study combined mechanical testing and plantar pressure measurements to validate the efficacy of density-graded midsoles. The methodology introduced here is relevant to realistic walking conditions, ensured by biomechanical tests supplemented by digital image correlation analyses. An optimization framework was then created to allocate foam densities at certain plantar zones based on the required cushioning performance constrained by the local pressure. The optimization algorithm was specifically tailored to accommodate varying local pressures experienced by different areas of the foot. The optimization strategy in this study aimed at reducing the overall weight of the midsole while ensuring there were no compromises in cushioning efficacy or distribution of plantar pressure. The approach presented herein has the potential to be applied to a wide range of gait speeds and user-specific plantar pressure patterns.