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The use of new mathematical modeling in the study of micro and Nano electro mechanical systems is currently becoming widespread. The scaling problem is apparent when the length of molecules, particles or grains immersed in the material is relatively important compared with the whole micro device dimension. Under this approach the classical theories of mechanics cannot describe suitably the structural requirement and it is necessary a more general outlook through non classical generalized theories which enclose the classical elasticity as a particular case where the non-classical constitutive parameters are negligible. When the microstructural effects are important, the material does not respond as a homogeneous but as a non-homogeneous one. A hundred years ago Cosserat brothers formulated a new theory of rigid grains which were embedded in an elastic macro medium; later Toupin, Mindlin along others researchers in 1960s developed a gradient strain theory which has been recently the source of many analystics and experimental investigations. In 1980s Ainfantis et al could develop a simplified strain gradient theory with just one additional non classical elastic constant which represents the volumetric elastic strain energy and characterized successfully the whole non classical pattern phenomenon. Beskos et al extended the treatment proposed initially by Aifantis and developed the first numerical applications for 2D and 3D boundary element methods and solved static as dynamic and crack problems. Since the times of Toupin and Mindlin it is looking for to establish a variational theory with a consistent cinematic and equilibrium boundary conditions, which seemed to have had success in the recent works of Amanatiodou and Aravas. This work presents the formulation of the hybrid boundary and finite element methods under the strain gradient scope which were developed by Dumont and Huamán through the versatile decomposition of the Hellinger-Reissner potential in two work principles: the displacements virtual work and the forces virtual work; both principles contain the virtual work performed by the non-classical magnitudes. Following, it is presented the complete development of singular and polynominal fundamental solutions abtained through the sixth order strain gradient differential equilibrium equations in terms of displacements. Next it is shown an application of the method to unidimensional truss element and bidimensional beam. Finally, it is presented a numerical application to strain gradient finite element, it is checked the patch tests to different elements orders and it is also shown a series of convergence analysis.