Three-dimensionally nonlocal tensile nanobars incorporating surface effect: A self-consistent variational and well-posed model

A naturally discrete nanobar implies that the continuum axiom is failed, and its surface-to-volume ratio is very large. The nonlocal theory of elasticity releasing the continuum axiom and the surface theory of elasticity are therefore employed to model tensile nanobars in this work. As commonly believed in the current practice, the axial nonlocal effect is only taken in account to analyze the mechanical behaviors of nanobars, regardless of their three-dimensional inherent atomistic interactions. In this study, a three-dimensional nonlocal constitutive law is developed to model the true nonlocal effect of nanobars, and based on which, a self-consistent variational bar model is proposed. It has been revealed for the first time how both the cross-sectional nonlocal interactions and the axial nonlocality affect the tensile behaviors of nanobars. It is found that the nonlocal influence predicted by the currently axial nonlocal bar model is grossly underestimated. Both the nonlocal cross-sectional and axial interactions become significant when the length-to-height ratio of nanobars is small. If the length-to-height ratio is relatively large (slender bars), the main nonlocal effect stems, however, from the nonlocal cross-sectional effect, rather than the axial nonlocal effect. This work also shows that it is possible to overcome the ill-posed problem of the pure nonlocal integral elasticity by employing both the pure nonlocal integral elasticity and surface elasticity. A well-posed size-dependent governing equation has been established for modeling nanobars under tension, and closed-form solutions are derived for their displacements. Based on the closed-form solutions, the effective elastic modulus is obtained and will be useful for calibrating the physical quantities in the discretecontinuum transition region for a span-scale modeling approach. It is shown that the effective elastic modulus may be softening or hardening, depending on the competition between the surface (modulus-hardening) and nonlocal (modulus-softening) effects.

Automated summary

This study demonstrates how both cross-sectional nonlocal interactions and axial nonlocality affect the tensile behaviors of nanobars. When the length-to-height ratio is small, nonlocal interactions become significant, while in slender bars, the main nonlocal effect stems from the nonlocal cross-sectional effect. Overcoming the ill-posed problem of pure nonlocal integral elasticity can be achieved by employing both pure nonlocal integral elasticity and surface elasticity.