Nonlinear Size-Dependent Analysis of Elastic Tiny-Bodies

Many researchers have studied the effect of surface energy on the
elastic behavior of nano-structural elements based on Gurtin and Murdoch surface
model. Many of them, however, assumed that the surface energy depends only on
the 2D symmetric infinitesimal surface strains and neglects the second-order
products of surface strains/displacement gradients. Moreover, there are some
researchers assumed that the surface energy is independent on infinitesimal
rotation tensor and neglected all rotationω
-related terms. These
previous descriptions of the surface stresses lack an important key point.
These descriptions assume a small strain deformation of the surface and neglect
the pre-strain that is, already, developed on the surface – before loading –
due the pre-tension stressσ0
. Consequently, considering the pre-strain, which is associated with the
deformation of the surface from the initial surface area to the deformed
年代urface area before loading, forces the surface to a state of large strain
after loading instead of small strain. Consequently, the accurate description
of surface elasticity is to consider the surface pre-strain, so a large strain
is assumed for surface strains while the bulk remain under small strain. This
forces us to exploit the Lagrangian surface description to consider the surface
pre-strain.