Exact continuum interpolation of the linear chain with hyper-pre-stress

In this paper we show that the second-gradient of strain model of Mindin interpolates exactly the reference configuration of the linear chain with hyper-pre-stress in the generic case when the ratio between the stiffnesses of near neighbors and next-to-near neighbors interactions is positive. This is possible only by including in the free-energy a null-lagrangian so that as a by-product we also relate null-lagrangians in the continuum description to the equilibrated force-systems associated to the geometric incompatibilities in the discrete setting. From a qualitative point of view, exact interpolation of the discrete model with hyper-pre-stress requires two internal lengths in the continuum and the model of Mindlin provide the minimal continuum frame for this. We obtain a one-parameter family of such interpolants and discuss in detail this non-generic situation.