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.