Stress function and its finite element implementation for elastostatic plain strain orthotropic problems of modified couple stress theory

• Published in: Acta mechanica 2024-05, Vol.235 (5), p.2725-2744
• Main Authors: Shang, Yan, Liu, Shi-Xuan, Cen, Song
• Format: Article
• Language: English
• Subjects: Aerospace engineering; Classical and Continuum Physics; Control; Deformation; Dynamical Systems; Elastostatics; Engineering; Engineering Fluid Dynamics; Engineering Thermodynamics; Equilibrium equations; Exact solutions; Finite element method; Heat and Mass Transfer; Mathematical analysis; Mechanics; Original Paper; Penalty function; Plane strain; Polynomials; Solid Mechanics; Stress functions; Theoretical and Applied Mechanics; Vibration
• ISSN: 0001-5970; 1619-6937
• DOI: 10.1007/s00707-024-03864-8

In this work, the stress function of the orthotropic plane strain problems in the modified couple stress theory (MCST) is first proposed and its polynomial analytical solutions are derived. Then, the stress function is used for developing a new 4-node membrane element for the MCST in that the C 1 continuity requirement is satisfied in weak sense using the penalty function method. In the element formulation, the stress function is adopted as the primary parameter for designing element’s stress and couple stress trial functions instead of assuming them directly. Therefore, the deduced stress and couple stress trial functions can satisfy both the equilibrium equations and deformation compatibility equations of the relevant problems a priori. Several numerical tests are examined and the results show that the element has good numerical accuracy and mesh distortion tolerance in simulating the size-dependent behaviors of small-scale orthotropic materials, proving that the usage of analytic trial function in the finite element implementation of the MCST can effectively improve element’s performance.