Finite element natural frequency analysis of cubic materials with couple stresses

Abstract A novel variational formulation is developed for natural frequency analysis of couple stress elastic continua. This formulation is based on extremization of a correlated action of the primary variables in the frequency domain. To reduce continuity requirements from $${C}^{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:math> to $${C}^{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msup> </mml:math> , Lagrange multipliers are employed to satisfy curvature–displacement compatibility and the essential boundary conditions on rotations in a variational sense, thus resulting in a mixed formulation with displacement and couple stress vectors as primary field variables. Using this variational framework to create a weak form, a two-dimensional finite element method is developed to carry out size-dependent natural frequency analysis of cubic elastic solids for the first time under consistent couple stress theory. The elements used are linear constant strain triangles with additional degrees of freedom on the edges, where tangential components of couple stress are assigned. Elements of this variety are referred to as edge elements or vector finite elements and have a distinct benefit in this case of ensuring the divergence free nature of couple stresses. Two fundamental computational examples are considered under plane strain conditions. First, a slender cantilever beam is examined for an isotropic material, along with corresponding single crystal aluminum and copper beams to explore the effects of anisotropy. A Rayleigh quotient analysis is carried out to quantify couple stress contributions to frequencies and mode shapes, and a modal participation analysis is performed to quantify how each mode contributes to motion in the coordinate directions. Finally, a hollow circular cylinder, both closed and with a small opening, is studied for the isotropic material and single crystal copper. Interesting results show a shift toward isotropic behavior for the copper cylinder and a reduction in the influence of the opening for both materials as the couple stress length parameter increases. With the ongoing push for new technologies on the micro- and nano-scale for applications in sensing, actuating and energy harvesting, the physical phenomena studied in this paper are important to consider for predicting mechanical behavior in this domain. In particular, understanding the role of size-dependent response becomes critical for effective design.