Beyond linearity: bent crystalline copper nanowires in the small-to-moderate regime

Several models can describe the nonlinear response of 1D objects to bending under a concentrated load. Successive stages consisting of geometrical and, additionally, mechanical non-linearity can be identified in moderately large extensions. We provide an explicit bending moment function with terms accounting for the linearity (Euler-Bernoulli), quasi-linearity, geometrical and finally, mechanical non-linearity as global features of a moderately large elastic deformation. We apply our method, also suitable for other metals, to the experimental data of Cu nanowires (NWs) with an aspect ratio of about 16 under different concentrated loadings. The spatial distribution of strain-hardening/softening along the wire or through the cross-section is also demonstrated. As a constitutive parameter, the strain-dependent stretch modulus represents, undoubtedly, changes in the material properties as the deformation progresses. At the highest load, the Green-Lagrange strain reaches a 12.5% extension with a corresponding ultra-high strength of about 7.45 GPa at the most strained volume still in the elastic regime. The determined stretch modulus indicates a significantly lower elastic response with an approximated Young's modulus (E ≅ 65 GPa) and a third-order elastic constant, C ₁₁₁ ≅ -350 GPa. Surprisingly, these constants suggest a 25-35% of that of the bulk counterparts. Ultimately, the method not only provides a quantitative description of the bent Cu NWs, but also indicates the robustness of the theory of nonlinear elasticity.