Longitudinal behavior of strain gradient bars

In this paper, the strain gradient theory, a non-classical continuum theory capable of capturing the size effect observed in micro-scale structures, is employed in order to investigate the size-dependent mechanical behavior of microbars. For a strain gradient bar, the governing equation of motion and classical and non-classical boundary conditions are derived using Hamilton's principle. Closed form solutions have been analytically obtained for static deformation, natural frequencies and mode shapes of strain gradient bars. The static deformation and natural frequencies of a clamped-clamped microbar subjected to a uniform axial distributed force are derived analytically and the results are depicted in some figures. The results indicate that contrary to the classical bars, strain gradient bars show size-dependent and stiffer mechanical behavior. In addition, a size-dependent strain gradient bar element has been developed in order to enable the finite element method (FEM) to numerically deal with the size-dependent problems in micro-scale structures where the attempts of the classical FEM have been in vain. The shape functions as well as the mass and the stiffness matrices are derived analytically based on Galerkin's method. During some examples, it is indicated that how the new element can be used in a problem and the results are compared to the analytical strain gradient results as well as the classical FEM and analytical results. A good agreement is observed between the strain gradient FEM and analytical results whereas the error of using the classical bar element is considerable. (C) 2013 Elsevier Ltd. All rights reserved.