Lattice Model with Nearest-Neighbor and Next-Nearest-Neighbor Interactions for Gradient Elasticity

Lattice models for the second-order strain-gradient models of elasticity theory are discussed. To combine the advantageous properties of two classes of second-gradient models, we suggest a new lattice model that can be considered as a discrete microstructural basis for gradient continuum models. It was proved that two classes of the second-gradient models (with positive and negative sign in front the gradient) can have a general lattice model as a microstructural basis. To obtain the second-gradient continuum models we consider a lattice model with the nearest-neighbor and next-nearest-neighbor interactions with two different coupling constants. The suggested lattice model gives unified description of the second-gradient models with positive and negative signs of the strain gradient terms. The sign in front the gradient is determined by the relation of the coupling constants of the nearest-neighbor and next-nearest-neighbor interactions.