Strain and velocity gradient theory for higher-order shear deformable beams

The strain and velocity gradient framework is formulated for the third-order shear deformable beam theory. A variational approach is applied to determine the governing equations together with initial and boundary conditions. Within the gradient framework, the strain energy is generalized to include strain as well as strain gradient. Furthermore, the kinetic energy is also generalized to include velocity and the velocity gradient. Such approach results in the introduction of the static and kinetic internal length scales. For dynamic analysis of beams, most of the gradient theories do not take the velocity gradient into account. The model developed in this paper depicts the influence of the velocity gradient on the governing equations and initial and boundary conditions of the third-order shear deformable theory. Through the assumption of the velocity gradients, kinematic quantities are distinguished on the microscale and on the macroscale. Finally, Timoshenko and Euler–Bernoulli beam theories are also presented by simplifying the third-order theory.