Large Elaso-plastic Deflection of Micro-beams Using Strain Gradient Plasticity Theory

Metallic micro-scale structures are observed to display more stiffness unlike the predictions of the classical plasticity theories, thus showing size-dependency. Hence, for accurate predictions of the behavior of micro-scale structures, non-classical theories are developed with intrinsic material length in their constitutive relation. Out of the many developed models, strain gradient plasticity theory is the one which is extensively followed. Power law based moment curvature relationship obtained from strain gradient plasticity is available in literature which models rigid-plastic behavior of micro-beam bending. In this article, elastic deformation is appended to the power law based moment curvature relationship. The derived law is employed to obtain the governing differential equation of a micro-cantilever under a normal terminal follower load undergoing large deflection. The non-linear differential equation is then solved by employing an efficient semi-incremental approach. In this approach the equation is solved by using Runge-Kutta 4th order method considering load to be at the current value but the material modulus at its previous value. Adoption of such a technique gave considerable computational advantage over the completely incremental method. The elasto-plastic force-displacement response curves of micro-beams showed stiffer results as compared to their classical counterpart. Also the elasto-plastic divergence from their corresponding elastic response curves are observed to happen at the same displacement for different micro-beams with same length to thickness ratio.