A penalty-based finite element framework for couple stress elasticity

Length scale based continuum mechanics can account for size dependence - a feature absent in classical theory that is becoming increasingly important in analyzing behavior of matter in micro- and nano-technology. Starting from a foundational mechanics and thermodynamics perspective, recent work has provided a self-consistent formulation to resolve certain issues associated with previous couple stress theory. In the current paper, a penalty-based finite element framework is developed to enable solution of general problems for linear elastic isotropic materials under plane strain conditions within this consistent couple stress theory (C-CST). The penalty parameter is incorporated in the formulation to satisfy continuity requirements, while allowing convergence of the method with only C0 elements. The efficacy of the finite element analysis is verified by studying three example problems for which closed form solutions are known. Furthermore, a fourth example problem for which a closed form solution is unknown is solved numerically to bring out nontrivial features of couple stress theory. The main objective of this work is to demonstrate that a simple finite element analysis-based framework can be effective in exploring the interesting features of C-CST. Additionally, detailed finite element analysis results provided in this work can be used to benchmark future computational development for size-dependent mechanics theory.