Effective behavior of lattices with angular interactions

 Angular interactions are of primary importance in mechanical trusses that they stabilize as well as in atomistic lattices, see Allinger and Tersoff-Brenner potentials. Graphenes are nowadays the best known example of hexagonal lattices. We will concentrate on the behavior of 2d-lattices undergoing deformations in the 3d-space, where major difficulties are already present when seeking for an equivalent behavior. We will give an example where homogenization is not required in the formulation of an equivalent continuous problem. We will show that for hexagonal lattices, on the contrary, homogenization is required even when only bond energy is taken into account. When angular interactions are added, we characterize the equivalent behavior by an alternate method. We will discuss the practical interest of the representation formulas.