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 succesful 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. • Implemented all the suggested amendments and improvements. • Stressed the importance of a double series expansions to investigate both non-trivial fundamental paths and non-trivial contributions of non-local elastic fractions in a two-phase linear elastic local/non-local mixture. • Commented upon the adherence of the results found in closed form with those provided by the literature, stressing the fact that we did not need using the ambiguous constitutive boundary conditions. • Provided graphical comparisons to show the capabilities of this technique to find interesting results also from the point of view of applications by means of a hierarchy of well-posed, easily solvable one-dimensional problems. • Shown how a well-known technique of matrix analysis may solve incremental problems to find the desired closed-form solutions. • Shown how a check by the exact solutions of inner balance may provide a definite validation of our approach. • Commented on the reliability of a limited expansion, with a hint on possible questions of (weak) convergence of the expansion.