Closed form solution for a nonlocal elastic bar in tension

A simple mechanical one-dimensional problem in the context of nonlocal (integral) elasticity is solved analytically. The nonlocal elastic material behaviour is described by the "Eringen model" whose nonlocality features all reside in the constitutive relation. This relation, of integral type, contains an attenuation function (usually assumed symmetric) aimed to capture the diffusion process of the nonlocality effects; it also exhibits a convolution format. The governing equation is a Fredholm integral equation of second kind whose analytical treatment, even for the usual choice of a symmetric kernel, is not easy to develop. In the present paper, assuming a specific shape for the attenuation function, a closed form solution in terms of strains is alternatively obtained by solving a Volterra integral equation of second kind. The latter can be easily solved with standard techniques, at least for the adopted kernel, taking also advantage from the symmetry of the solution. Such a closed form solution is an essential result to validate the effectiveness of numerical procedures aimed to solve more complex mechanical problems in the context of nonlocal elasticity.