Gradient elasticity and size effect for the borehole problem

Abstract

It has been shown recently that robust gradient models of elasticity offer an effective tool to resolve existing difficulties (e.g., elimination of singularities) or predict new experimentally observed phenomena (e.g., size effects) not captured by classical theory. The price that one pays for it, however, is the need to determine extra phenomenological coefficients and invent new boundary conditions associated with the gradient terms. The modest contribution of this note is to show that even the simplest possible gradient elasticity model may yield entirely different results for different higher-order boundary conditions used. This is demonstrated by considering the borehole problem, i.e., the determination of the state of stress and strain in an externally loaded or internally pressurized body containing a cylindrical hole. The standard practice of using variationally consistent boundary conditions leads to difficulties in physically interpreting them as well as in complex solution formulae without much physical insight. Alternative, mathematically less elegant arguments, as those employed here, lead to much neater formulas, usually consisting of a sum of the classical elasticity solution and an extra gradient term. These may easily be utilized to address engineering applications, ranging from processing/fabrication of metallic specimens at the micron and nanoscales to mining/drilling operations and tunneling excavation at geo-scales. The results are by no means conclusive, only suggestive of the various possibilities and choices that one can make at present. This, in itself, points to the pressing need of further work on the issue of extra boundary conditions with the aid of new novel experiments and accompanied multiscale simulations.