Nonlocal Finite Element Method For Bending Of Beams Using Eringen’s Integral Formulation

Nonlocal theory of elasticity is different from the classical local one, whose nonlocality features contains an attention function aimed to capture the diffusion phenomenon of the nonlocality effects. Take full account of scale effect and internal microstructure effect will be better to explain the micro problems, which has a very high academic value.The integral form constitutive equation is the basis of Eringen nonlocal theory, while the complex integral differential equation makes it difficult to get the analytical solution, and the whole domain integral also increases more complex to the traditional numerical method.Based on Eringen’s integral formulation, the total potential energy is deduced and a finite element method of nonlocal integral elasticity is established to solve bending of Euler-Bernoulli beams. Since the nonlocal material behavior is vanishing beyond an influence distance, a limited influence distance is provided to simplify the integral computation. A nonlocal finite element programming is implemented and tested. And some typical examples such as simply supported beam and cantilever beam are analysed using the nonlocal FE code. The influence of internal length, phase parameter, element length and mesh density, are investigated in details to show the nonlocality features.