Non-affinity in multi-material mechanical metamaterials

Non-affine deformations enable mechanical metamaterials to achieve their unusual properties while imposing implications for their structural integrity. The presence of multiple phases with different mechanical properties results in additional non-affinity of the deformations, a phenomenon that has never been studied before in the area of extremal mechanical metamaterials. Here, we studied the degree of non-affinity, Gamma, resulting from the random substitution of a fraction of the struts, rho(h), that make up a lattice structure and are printed using a soft material (elastic modulus = E-s) by those printed using a hard material (E-h). Depending on the unit cell angle (i.e., theta = 60 degrees, 90 degrees, or 120 degrees), the lattice structures exhibited negative, near-zero, or positive values of the Poisson's ratio, respectively. We found that the auxetic structures exhibit the highest levels of non-affinity, followed by the structures with positive and near-zero values of the Poisson's ratio. We also observed an increase in Gamma with E-h/E-s and rho(h) until E-h/E-s=10(4) and rho(h)=75%-90% after which Gamma saturated. The dependency of Gamma upon rho(h) was therefore found to be highly asymmetric. The positive and negative values of the Poisson's ratio were strongly correlated with Gamma. Interestingly, achieving extremely high or extremely low values of the Poisson's ratio required highly affine deformations. In conclusion, our results clearly show the importance of considering non-affinity when trying to achieve a specific set of mechanical properties and underscore the structural integrity implications in multi-material mechanical metamaterials.