Since composite materials have been developed, many types of materials (e.g. carbon
fibre, carbon nanotubes (CNTs)) can be embedded in a standard matrix in order to
obtain materials with enhanced physical properties. To investigate enhanced properties
of nano-composites, not only mechanical properties but also electrical properties
should be taken into account. Furthermore, at the nano-scale, covalent forces between
atoms play a crucial role in their behaviour. This thesis is focused on eletromechanical
effects (i.e. piezoelectricity and flexoelectricity) and the size effect in micro/nano materials.
The aim is to implement continuum modelling solutions for nonlocal/gradient
elastic problems in which size effect plays a significant role in material behaviour. The
FEniCS Project is used to provide a novel tool for automated solutions of partial differential
equations (PDE) by the finite element method. In particular, it offers significant
flexibility with regards to discretization choices for triangular elements. When implementing
a nonlocal/strain gradient elastic framework using FEniCS, a weak form of the
gradient elasticity derived from the Principal of Virtual Work (PVW) is required. Due
to the fourth order PDE in term of displacements in the gradient elasticity, C1 continuous
elements (e.g. Hermitian finite element) are usually required. However, to avoid
the use of C1 continuous elements, an equivalent mixed-type finite element formulation
is considered. To investigate the material behaviour, strain gradient finite element
formulations based on a mixed variational approach are used. Numerical results are
compared with analytical solutions or experimental data to confirm the convergence
and accuracy of the simulations. To extend the capability of the implementation to
allow the modelling of nanocomposites efficiently, Extended Finite Element Method
(XFEM) is introduced. By increasing mesh density only around the discontinuities,
the resulting program runs faster than if a finer mesh had been used everywhere, with
the additional benefit that more accurate results are obtained.