A perturbation approach to two-phases, non-local, fully deformable beams

The local theory of elasticity (inner forces are sensible at insensible intermolecular distances) faces inconsistencies and limitations when one considers bodies at very small scales, i.e., with characteristic dimensions that are not several orders of magnitude greater than the intermolecular lengths, even in a linear setting. The so-called quasi-continuum models, preserving the principles of kinematics and balance of ordinary continuum mechanics while incorporating a richer description of inner forces at the constitutive level, attempt to mitigate this issue. One such model, well-known and commonly adopted in the last years, is due to Eringen and linearly expresses stress in terms of strain in a differential or integral form, by resorting to the convolution of a kernel function. This model, while successful for infinite media, encounters possible drawbacks when applied to finite domains, necessitating the imposition of “constitutive boundary conditions” of uncertain physical meaning. A series of alternative proposals in the literature try to overcome such difficulty; in the present contribution, we apply a perturbation procedure that circumvents this requirement. We apply this methodology to analyse paradigmatic problems of statics and free dynamics for fully deformable beams, and we present closed-form first-order expressions for benchmark scenarios, avoiding the necessity to use the constitutive boundary conditions. The solutions for purely flexible, Bernoulli–Euler, beams can be attained as a particular case of those provided here.