The mathematical modeling of micro-devices in which structure and the microstructure are about the same scale of magnitude, as well as of macrostructure of markedly granular or crystal nature (microcomposites), demands a nonlocal approach for strains and stresses. More than one hundred years ago the Cosserat brothers had already developed a theory for rigid grains. However, and in no detriment due to Toupin and other researchers, Mindlin s work in the 1960s may be accounted the basis of the so-called strain gradient theory, which has recently become the subject of a large number of analytical and experimental investigations motivated by the development of news structural materials together with the increasing use of micro and nano-mechanical devices in the industry. More recently, Aifantis and coworkers managed to develop a simplified strain gradient theory based only on two additional elasticity constants that are representative of material lengths related to surface and volumetric strain energy. A series of very recent works done by Beskos and collaborators extended the field of applications of Aifantis propositions and introduced a fundamental solution that actually remounts to developments already laid down by Mindlin. Beskos workgroup may be regarded as the proponent of the first of the first boundary element 2D and 3D implementations on the subject for both statics and frequency-domain analyses, also including crack problems. Since Toupin and Mindlin`s time, investigations have been under development to establish the variational basis of the theory and to consistently formulate equilibrium and kinematic boundary conditions established by Amanatidou and Aravas. This dissertation makes a revision of the gradient strain elasticity theory and presents a didactic study of the simplest problem that can be conceived, i.e., a bar under different axial actions (Aifantis, Beskos). The fundamental solution for 2D and 3D problems is also presented and studied for an elastic medium submitted to a point force, for boundary methods developments, as well as submitted to polynomial stress fields (for static problems), as in the hybrid finite element method. It is shown that Aifantis strain gradient theory may be developed in the context of the Hellinger-Reissner potential, for the sake of hybrid finite and boundary element implementations. Goal of the present research work is as a detailed study of state art of the theme, which comprises an investigation of the singular integrals one must deal with in a computational implementation. The complete computational development for static and dynamic hybrid boundary/finite analyses is planned for a future doctoral thesis.