Comprehensive and easy-to-use torsion and bending theories for micropolar beams

• Classical Duleau torsion and Timoshenko bending theories of beams are generalized.
• Easy-to-use torsion and bending theories for micropolar beams are developed/solved.
• Useful for beams with different cross-sections, boundary conditions, and loading.
• Microstructure-dependent size effects and the microinertia effects are captured.
• A finite element method for solving the dynamic equations is presented and used.

The main goal of this paper is to develop a comprehensive beam model based on the micropolar elasticity theory which is as general, as easy to use, and as convenient as the classical beam theories. Uncomplicated torsion and bending theories for micropolar elastic beams deforming in three-dimensional space and under different types of external loading and boundary conditions are presented in this paper. Unlike the classical beam models, the developed beam model includes the effect of microinertia and contains new material parameters to capture the microstructure-dependent size effects which could be useful when dealing with micro scale beams. The presented micropolar beam model generalizes the Duleau torsion and Timoshenko bending beam models to include the microstructure effects. Hamilton's principle and a variational approach are used to derive the dynamic equations of the micropolar beam with longitudinal, torsional, and bending deformations. Then the governing dynamic equations are solved numerically by using a finite element approach and numerical results for a simply supported micropolar beam are provided. The static and dynamic behaviors of the developed micropolar beam model are studied and compared against the classical beam models. In particular, the conditions for recovery of the results of the classical beam theories, i.e. Duleau and Timoshenko theories, are addressed.