Iterative methods for nonlocal elasticity problems

Abstract

Nonlocal elasticity is addressed in the context of geometrically linearised structural models with linear and symmetric constitutive relations between dual fields, with physical interpretation of stress and elastic strain. The theory applies equally well to formulations as integral convolutions with an averaging symmetric kernel and to those involving an elastic potential depending on local gradients. According to the original strain-driven nonlocal elastic model, input and output of the stiffness operator are, respectively, the elastic strain field and the stress field. The swapped correspondence is assumed in the recently proposed stress-driven nonlocal elastic model, with input and output of the compliance operator, respectively, given by the stress field and the elastic strain field. Two distinct nonlocal elasticity models, not the inverse of one another, may thus be considered. The strain-driven model leads to nonlocal elastostatic problems which may not admit solution due to conflicting requirements imposed on the stress field by constitutive law and equilibrium. To overcome this obstruction, several modifications of the original scheme have been proposed, including mixtures of local/nonlocal elastic laws and compensation of boundary effects. On the other hand, the stress-driven model leads to nonlocal elastostatic problems which are consistent and well-posed. Two iterative methods of solution, respectively, for stress-driven and mixture strain-driven models, are here contributed and analysed. The relevant algorithms require only solutions of standard local elastostatic problems. Fixed points of the algorithms are shown to be coincident with solutions of the pertinent nonlocal elastostatic problem. It is also proven that, for statically determinate structural models, the iterative method pertaining to the nonlocal stress-driven model converges to the solution just at the first step. For statically indeterminate ones, the computations reveal that convergence is asymptotic but very fast. Clamped-free and clamped-supported nano-beams are investigated as standard examples in order to provide evidence of the theoretical results and to test and show performance of the iterative procedures.