Using the total energy balance equation and principle of the frame indifference, a
fundamental set of relations of the local gradient continuum model of elastic solids is
formulated. The model is based on taking account of non-convective and
non-diffusive mass flux related to the changes in the material microstructure. Linear
stationary governing equations of the local gradient theory and corresponding
boundary conditions are also derived by variational principle. In order to investigate
the size-dependent behavior of nano-scale structures, this model is combined with the
Bernoulli–Euler beam theory. Deflection of the cantilever beam subjected to the
end-point loading under the plane stress conditions is evaluated and compared to the
corresponding ones provided by the classical theory and by the strain gradient theory.
It is shown that the beam deflection within the local gradient theory is smaller than
that predicted by the classical Bernoulli–Euler beam theory. This work may be
of special interest for designing the devices utilizing the micro/nano-beam
elements.
Institute of Construction and
Architecture
Slovak Academy of Sciences
84503 Bratislava 45
Slovakia
Center of Mathematical
Modeling
Pidstryhach Institute for Applied Problems of Mechanics and
Mathematics
National Academy of Sciences of Ukraine
79005 Lviv
Ukraine
