| Numerous experiments conducted on polymers by various researchers have shown the presence of length scale dependent deformation at micron and submicron length scales. Such length scale dependent deformation strongly affects the mechanical properties of materials. For instance, micro beam bending experiments have revealed an increase in the bending stiffness with decreasing beam thickness. Similarly, increase in the hardness with decreasing probing depth has been observed in indentation type testing.;This length scale dependent behavior cannot be captured by classical (local) continuum theories as they contain no 'intrinsic material length scales' and consequently predict no length scale effects. Material formulations incorporating higher order gradients of displacements have been successful in modeling the effects of such size dependent behavior. However, a direct implementation of such higher order displacement gradient theories into finite element is challenging as these theories consist of fourth order partial differential equations and would require both displacements and their first derivatives to be continuous over the element boundaries, which consequently results in a large number of degrees of freedom.;Based on the couple stress strain gradient elasticity theory, a novel variational approach with a penalty term is suggested in this work to develop a simple finite element formulation with fewer degrees of freedom. In this framework, rotations are introduced as independent nodal variables along with nodal displacements, and the differences between the rotational fields (determined by the displacements) and rotation degrees of freedom are minimized by a penalty term.;On the basis of the proposed approach, numerical simulations are performed to study the length scale dependent deformation in bending of micro-beams and the results are compared with available experimental data in the literature. Comparison of these simulations with the experimental results demonstrates that the suggested approach has the capability of modeling such size effects at small length scales. It is also shown that the proposed approach is successful for the interpretation of probing depth dependent deformation behavior in hardness when using sharp and spherical indenters. |
