Non-affinity in multi-material mechanical metamaterials
Mohammad J. Mirzaali, H. Pahlavani, Ebrahim Yarali, Amir A. Zadpoor
Abstract
Abstract 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, $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Γ</mml:mi></mml:math> , resulting from the random substitution of a fraction of the struts, $${\rho }_{h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ρ</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math> , that make up a lattice structure and are printed using a soft material (elastic modulus = $${E}_{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math> ) by those printed using a hard material ( $${E}_{h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math> ). Depending on the unit cell angle (i.e., $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>θ</mml:mi></mml:math> = 60°, 90°, or 120°), 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 $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Γ</mml:mi></mml:math> with $$\frac{{E}_{h}}{{E}_{s}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:msub><mml:mi>E</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:msub><mml:mi>E</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mfrac></mml:math> and $${\rho }_{h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ρ</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math> until $$\frac{{E}_{h}}{{E}_{s}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:msub><mml:mi>E</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:msub><mml:mi>E</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mfrac></mml:math> =10 4 and $${\rho }_{h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ρ</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math> = 75%-90% after which $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Γ</mml:mi></mml:math> saturated. The dependency of $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Γ</mml:mi></mml:math> upon $${\rho }_{h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ρ</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math> was therefore found to be highly asymmetric. The positive and negative values of the Poisson’s ratio were strongly correlated with $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Γ</mml:mi></mml:math> . 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.