Nonlocal Elasticity Problems Research Articles

2D nonlocal elasticity: Investigation of stress and strain fields in complex shape regions

AbstractThe paper presents the features of solutions to nonlocal elasticity problems in domains with stepped transitions and elliptical notches. A comparative analysis of the solutions obtained in the classical and nonlocal elasticity theories was carried out, as well as an analysis of the influence of the main parameters of the model on the final results. The presented solutions for the region with a step transition are in qualitative agreement with the experimental data and have similar portraits of deformations and resultant stresses along the loading axis. In regions with elliptical notches, similar effects are observed. In addition to qualitative agreement with experiments, solutions in the nonlocal formulation have lower stress values in the concentrators and exhibit edge effects, which are characterized by sharp stress and strain gradients at the domain boundaries. The solutions were carried out numerically by the finite element method.

On formulation of nonlocal elasticity problems

Nonlocal elasticity models are tackled with a general formulation in terms of source and target fields belonging to dual Hilbert spaces. The analysis is declaredly focused on small movements, so that a geometrically linearised approximation is assumed to be feasible. A linear, symmetric and positive definite relation between dual fields, with the physical interpretation of stress and elastic states, is assumed for the local elastic law which is thus governed by a strictly convex, quadratic energy functional. Genesis and developments of most referenced theoretical models of nonlocal elasticity are then illustrated and commented upon. The purpose is to enlighten main assumptions, to detect comparative merits and limitations of the nonlocal models and to focus on still open problems. Integral convolutions with symmetric averaging kernels, according to both strain-driven and stress-driven perspectives, homogeneous and non-homogeneous elasticity models, together with stress gradient, strain gradient, peridynamic models and nonlocal interactions between beams and elastic foundations, are included in the analysis.

Iterative methods for nonlocal elasticity problems

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.

Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams

Abstract In the strain-driven model of nonlocal elasticity proposed by Eringen , the elastic strain is defined by a Fredholm integral equation in which the stress is the output of a convolution between the local response to an elastic strain and a smoothing kernel dependent on a nonlocal parameter. In the wake of this proposal, size effects in nano-beams were investigated in literature by adopting a differential formulation considered to be equivalent to the integral one. Recent improvements have however revealed that equivalence requires also the fulfilment of constitutive boundary conditions. Moreover, this strain-driven nonlocal elastic problem has been shown to be ill-posed, being conflicting with equilibrium requirements. A stress-driven integral constitutive law provides the natural way to get well-posed nonlocal elastic problems for application to nano-structures. The new integral constitutive law is formulated with explicit reference to plane and straight nano-beams according to the standard Bernoulli - Euler structural model. The solution procedure based on the stress-driven nonlocal law is described and adopted for the solution of a simple statically indeterminate scheme, thus showing effectiveness of the new model for the structural design of nano-devices.

Benchmarks in nonlocal elasticity defined by Eringen’s integral model

In this paper we present low-residual approximate solutions for nonlocal 1D and 2D elasticity problems defined according to Eringen’s integral model. The benchmarks in the 1D cases are defined by prescribing the stress field while the unknown fields are the strains or the displacements. For the 2D cases we define problems with equilibrated tractions and evaluate the approximate displacement field. Meanwhile a Fourier series as well as a set of Chebyshev polynomials are used as the basis functions for the main unknown fields. We increase the number of the approximation functions to decrease the norm of the residuals and repeat the procedure until reasonable accuracy is obtained for the final solution. Since the procedure is very time consuming, in some benchmark problems we present the calculated coefficients and in some other we give some point-wise values for further use. The results presented in this paper are particularly useful for the validation and convergence studies when numerical methods are to be used for the solution of the nonlocal elasticity problems.

Sensitivity filtering from a continuum mechanics perspective

In topology optimization filtering is a popular approach for preventing numerical instabilities. This short note shows that the well-known sensitivity filtering technique, that prevents checkerboards and ensures mesh-independent designs in density-based topology optimization, is equivalent to minimizing compliance for nonlocal elasticity problems known from continuum mechanics. Hence, the note resolves the long-standing quest for finding an explanation and physical motivation for the sensitivity filter.