Closed-form frequency solutions for simplified strain gradient beams with higher-order inertia

• Euler−Bernoulli beam model with strain gradient parameters is developed.
• The possible boundary conditions of different end constraints are discussed.
• Analytical solutions for the proposed boundary value problems are obtained.
• Effects of gradient parameters, Poisson ratio and boundary conditions are studied.

This paper develops an Euler−Bernoulli beam model within the context of a simplified strain gradient theory with higher-order inertia. In contrast to the classical beam models, the proposed gradient beam models can capture the size effects by introducing not only the internal length l2 related to strain gradient but also the internal length l1 related to velocity gradient. The governing equation of motion and boundary conditions are derived by using the variational principles. The closed-form solutions for free vibrations of beams with three typical boundary conditions are obtained. Numerical results show that the choices of the higher-order boundary conditions have a minor effect on the natural frequencies of beams. In addition, the inclusion of the strain gradient parameter l2 increases the effective stiffness of beams and hence it increases the natural frequencies of beams; whereas the inclusion of the velocity gradient parameter l1 acts as an equivalent compression force in the governing equation and therefore it leads to the decrease of the natural frequencies of beams. Moreover, the significant Poisson effect on the natural frequencies of beams is observed when the thickness of the beam is comparable to the strain gradient parameter l2. The closed-form solutions for natural frequencies of beams presented in this work may serve as benchmark results for other numerical methods.