A nonlinear strain gradient finite element for microbeams and microframes

Abstract

The aim of this paper is to develop a total Lagrangian finite element formulation for the geometrically nonlinear deformation analysis of microbeams and microframes. A general form of the first strain gradient elasticity theory of Mindlin is employed to capture size effects at micron scales. Moreover, the von Karman strain tensor and its spatial gradient are used to include the effects of geometric nonlinearity. Due to the higher-order nature of the gradient-based governing equilibrium equations, the interpolation functions of the extension and bending field variables are chosen to be the standard \(C^1\) and \(C^2\) functions, respectively. Accordingly, a novel two-node strain gradient microbeam element is introduced. The formulation is then extended to include the deformation of microbeams with arbitrary orientation in a plane, which enables us to analyze arbitrary planar microframes composed of several microbeams at different orientations. The full Newton–Raphson scheme is used to solve the resulting nonlinear algebraic equations. Three different examples are analyzed to show the accuracy and performance of the proposed element at linear as well as nonlinear regimes of deformation. It is shown that the new element can successfully capture the so-called size effect as well as geometric nonlinearity at large distortion of microbeams and microframes at micron scales.