A simplified moment-curvature based approach for large deflection analysis of micro-beams using the consistent couple stress theory

Abstract

Abstract Micro-scale structures are observed to be mechanically stiffer than the predictions obtained from the conventional elastic theories. This led to the development of models which include the effect of couple stresses to capture the size-dependent behavior. This paper develops a large elastic deflection version of the micro-beam analysis within the consistent couple stress theory framework. A material length parameter naturally appears in the moment-curvature relationship to account for the length scale effect. The method proposed here shows that starting with a moment-curvature relation unlike stress-strain relationship leads to a non-linear governing differential equation in terms of slope of the deformed beam. The differential equation is integrated to obtain the expressions for co-ordinates of the deformed beam which are in the form of improper integrals. An efficient semi-analytical approach involving removal of singularity is introduced. The proposed method is easy to implement as compared to the elliptic integral approach and is faster than direct numerical integration. Upon inclusion of couple stress effects, the static bending results show a stiffer response than the classical elasticity theory. It is also shown in this work that a common governing differential equation is obtained for predicting the deflections of micro as well as macro-beams.