Nonlocal or gradient elasticity macroscopic models: A question of concentrated or distributed microstructure

• Strain gradient models or stress gradient models are both shown to be supported by some microstructured arguments.
• A continualization approach is applied to the governing difference equations.
• Nonlocal models are built from concentrated microstructure.
• Gradient elasticity models are associated to some distributed microstructure.
• Local neighboring interaction may transmit some higher-order effects at the macroscopic scale.

One dimensional discrete systems, such as axial lattices, may be investigated by using some enriched continuum models. In this paper, strain gradient models (also called gradient elasticity) and stress gradient models (also called nonlocal elasticity) are both shown to be supported by some microstructured physical configurations. Starting from the difference equations associated with each discrete system, a continualization approach is applied to the governing difference equations. Alternatively, one may use energy considerations to derive these higher-order continua. Stress gradient models are built from concentrated microstructure (with direct neighboring interaction) whereas strain gradient models are associated to some distributed microstructure (also with direct neighboring interaction). Each model leads to opposite effect, namely the softening effect of the small scale terms for the stress gradient model (built from concentrated microstructure), and the stiffening effect of the small scale terms for the strain gradient model (built from distributed microstructure) with respect to the asymptotic local model. We also discuss the link between lattice equations, finite difference formulation or finite element formulation of the continuous local problem. The paper concludes that the local neighboring interaction at the discrete scale may transmit some higher-order effects at the macroscopic scale. Hence, the higher-order nature of the macroscopic constitutive laws may not necessarily be seen as the consequence of nonlocal interaction at the lattice scale.