Temperature-dependent multi-scale modeling of surface effects on nano-materials

In this paper, a novel temperature-dependent multi-scale method is developed to investigate the role of temperature on surface effects in the analysis of nano-scale materials. In order to evaluate the temperature effect in the micro-scale (atomic) level, the temperature related Cauchy–Born hypothesis is implemented by employing the Helmholtz free energy, as the energy density of equivalent continua relating to the inter-atomic potential. The multi-scale technique is applied in atomistic level (nano-scale) to exhibit the temperature related characteristics. The first Piola–Kirchhoff stress and tangential stiffness tensor are computed, as the first and second derivatives of the free energy density to the deformation gradient, which are transferred to the macro-scale level. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo-mechanical finite element model, and an intrinsic function is employed to model the surface and temperature effects in macro-scale level. The stress and tangential stiffness tensors are derived at each quadrature point by interpolating the data from nearby representative atom. The boundary Cauchy–Born (BCB) elements are introduced to capture the surface, edge and corner effects. Finally, the numerical simulation of proposed model together with the direct comparison with fully atomistic model illustrates that the technique provides promising results for facile modeling of boundary effect on thermo-mechanical behavior of metallic nano-scale devices.