Mixed displacement and couple stress finite element method for anisotropic centrosymmetric materials

• Published in: European journal of mechanics, A, Solids A, Solids, 2021-01, Vol.85, p.104074, Article 104074
• Main Authors: Pedgaonkar, Akhilesh, Darrall, Bradley T., Dargush, Gary F.
• Format: Article
• Language: English
• Subjects: Anisotropic materials; Centrosymmetric materials; Consistent couple stress theory; Constitutive relationships; Curvature; Displacement; Elasticity; Finite element analysis; Finite element method; Mixed variational formulation; Strain; Stress concentration; Symmetry; Tensors
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2020.104074

The classical theory of elasticity is an idealized model of a continuum, which works well for many engineering applications. However, with careful experiments one finds that it may fail in describing behavior in fatigue, at small scales and in structures having high stress concentration factors. Many size-dependent theories have been developed to capture these effects, one of which is the consistent couple stress theory. In this theory, couple stress μij is present in addition to force stress σij and its tensor form is shown to have skew symmetry. The mean curvature κij, which is defined as the skew-symmetric part of the gradient of rotations, is the correct energy conjugate of the couple stress. This mean curvature κij and strain eij together contribute to the elastic energy. The scope of this paper is to extend the work to study anisotropic materials and present a corresponding finite element method. A fully displacement based finite element method for couple stress elasticity requires C1 continuity. To avoid this, a mixed formulation is presented with primary variables of displacements ui and couple stress μi vectors, both of which require only C0 continuity. Centrosymmetric classes of materials are considered here for which force stress and strain are decoupled from couple stress and mean curvature in the constitutive relations. Details regarding the numerical implementation are discussed and the effect of couple stress elasticity on anisotropic materials is examined through several computational examples. •Novel C0 mixed finite element method is developed for consistent couple stress theory.•Weak form is written in terms of displacement and couple-stress polar vectors.•General method is applied to anisotropic centrosymmetric materials for first time.•Examples consider cubic, hexagonal, trigonal, tetragonal single crystal materials.•Stress concentration factor and non-dimensional stiffness studied in model problems.