Size-dependent couple stress natural frequency analysis via a displacement-based variational method for two- and three-dimensional problems

After deriving a novel Principle of Stationary Correlated Action for the general bi-anisotropic case of size-dependent materials governed by consistent couple stress theory, a Ritz Spline Method is created for free vibration analysis. The underlying formulation is then specialized for isotropic materials requiring the specification of only a single couple stress material parameter that can be defined in terms of an effective intrinsic material length scale. The framework permits both two- and three-dimensional analysis using a multivariate B-spline basis that enables the flexibility to select a range of piecewise polynomial orders and higher levels of continuity, as required in couple stress theory. To validate the formulation and numerical implementation, two example problems involving a cantilever with rectangular cross section and a free-free cylindrical tube are studied in detail. Interestingly, one finds that while couple stress effects tend to stiffen the structures, the effect is not uniform. The natural frequencies of some modes are significant enhanced as the couple stress effective length scale increases, while other modes are nearly or completely unaffected. In addition, the important three-dimensional nature of several aspects of consistent couple stress continuum mechanics is clearly observed from these two basic problems of free vibration.