Analytical and numerical procedures for a nonlocal elastic 1D problem

An analytical and two numerical solutions, obtained via FEM-based techniques, for a simple mechanical one-dimensional problem, namely a bar in tension, and in the context of nonlocal (integral) elasticity, are discussed in this paper. To describe the nonlocal material behavior the “Eringen model” is adopted. This model incorporates all the nonlocality features of the material in the constitutive relation by means of an attenuation function which is devoted to the description of the diffusion processes arising in a nonlocal medium. With this constitutive model the governing equation pertaining the simple 1D problem studied is a Fredholm integral equation of second kind whose analytical treatment exhibits, in general, some difficulties. In this paper, for a specific attenuation function, a closed form solution of the problem, in terms of strain, is given by alternatively solving a Volterra integral equation of second kind. Such a closed form solution is then utilized as a reference solution to validate two different FE procedures, first proposed in [1], and here illustrated with reference to the addressed 1D nonlocal mechanical problem.