Micro-to-macro transitions for continua with surface structure at the microscale

A geometrically non-linear framework for micro-to-macro transitions is developed that accounts for the effect of size at the microscopic scale. This is done by endowing the surfaces of the microscopic features with their own (energetic) structure using the theory of surface elasticity. Following a standard first-order ansatz on the microscopic motion in terms of the macroscopic deformation gradient, a Hill-type averaging condition is used to link the two scales. The surface elasticity theory introduces two additional microscopic length scales: the ratio of the bulk volume to the energetic surface area, and the ratio of the surface and bulk Helmholtz energies. The influence of these microscopic length scales is elucidated via a series of numerical examples performed using the finite element method.