Strain gradient formulation of functionally graded nonlinear beams

In this paper size-dependent static and dynamic behavior of nonlinear Euler–Bernoulli beams made of functionally graded materials (FGMs) is investigated on the basis of the strain gradient theory. The volume fraction of the material constituents is assumed to be varying through the thickness of the beam based on a power law. As a consequence, the material properties of the microbeam (including length scales) are varying in the direction of the beam thickness. To develop the model, the usual simplifying assumption which considers the length scale parameter to be constant through the thickness is avoided and equivalent length scale parameters are introduced for functionally graded microbeams as functions of the constituents’ length scale parameters and volume fraction. Considering the mid-plane stretching that causes the nonlinearity in the beam behavior, the nonlinear governing equation and both classical and non-classical boundary conditions are obtained using Hamilton’s principle. General presented governing equation and the boundary conditions have been specialized for a hinged-hinged beam as a specific case and the static deflection and free vibration of the FG hinged-hinged microbeam are investigated. The results of the nonlinear strain gradient theory are compared with those calculated based on the linear strain gradient theory, linear and nonlinear modified couple stress theory, and also the linear and non-linear classical models, noting that the couple stress and the classical theories are indeed special cases of the strain gradient theory.