The Reproducing Kernel Element Method (RKEM) implementation of the Fleck-Hutchinson phenomenological strain gradient theory in 1D, 2D and 3D is implemented in this research. Fleck-Hutchinson theory fits within the framework of Touplin- Mindlin theories and deals with first order strain gradients and associated work conjugate higher-order stress. Theories at the intrinsic or material length scales find applications in size dependent phenomena. In elasticity, length scale enters the constitutive equation through the elastic strain energy function which depends on both strain as well as the gradient of rotation and stress. The displacement formulation of the Touplin Mindlin theory involve diffrential equations of the fourth order, in conventional finite element method C1 elements are required to solve such equations, however C1 elements are cumbersome in 2D and unknown in 3D. The high computational cost and large number of degrees of freedom soon place such a formulation beyond the realm of practicality. Recently, some mixed and hybrid formulations have developed which require only C0 continuity but none of these elements solve complicated geometry problems in 2D and there is no problem yet solved in 3D. The large number of degrees of freedom is still inevitable for these formulation. As has been demonstrated earlier RKEM has the potential to solve higher-order problems, due to its global smoothness and interpolation properties. This method has the promise to solve important problems formulated with higher order derivatives, such as the strain gradient theory.

Nonlinear finite element models of functionally graded beams with power-law variation of material, accounting for the von-Karman geometric nonlinearity and temperature dependent material properties as well as microstructure dependent length scale have been developed using the Euler-Bernoulli as well as the first-order and third- order beam theories. To capture the size effect, a modified couple stress theory with one length scale parameter is used. Such theories play crucial role in predicting accurate deflections of micro- and nano-beam structures. A general third order beam theory for microstructure dependent beam has been developed for functionally graded beams for the first time using a modified couple stress theory with the von Karman nonlinear strain. Finite element models of the three beam theories have been developed. The thermo-mechanical coupling as well as the bending-stretching coupling play significant role in the deflection response. Numerical results are presented to show the effect of nonlinearity, power-law index, microstructural length scale, and boundary conditions on the bending response of beams under thermo-mechanical loads. In general, the effect of microstructural parameter is to stiffen the beam, while shear deformation has the effect of modeling more realistically as a flexible beam.