Dynamics of non-local systems handled by fractional calculus

Mechanical vibrations of non-local systems with long-range, cohesive, interactions between material particles have been studied in this paper by means of fractional calculus. Long-range cohesive forces between material particles have been included in equilibrium equations assuming interaction distance decay with order α . This approach yields as limiting case a partial fractional differential equation of order α involving space-time variables. It has been shown that the proposed model may be obtained by a discrete, mass-spring model that includes non-local interactions by non-adjacent particles and the mechanical vibrations of the particles have been obtained by an approximation fractional finite difference scheme already used for static analysis. Modal shapes and natural frequency of the non-local systems may then be obtained from the proposed model with boundary conditions coalescing with classical mechanics boundary conditions and solution obtained with the proposed model is capable to capture local characters as particular case of the real coefficient α . Numerical applications reported show a remarkable non-local feature of the state variables of the analyzed system.