A third-gradient 1D continuum obtained via asymptotic expansion from a micro-structure obtained constraining a sequence of modified Hart’s antiparallelograms

In this paper, we show one solution of the following synthesis problem: to find a planar, periodic, structure made up of straight bars linked by (perfect) hinges, which, once homogenized, can be modelled as a planar third-gradient one-dimensional (1D) continuum. One possible-solution structure is obtained by considering a suitably Modified Hart’s Antiparallelograms Mechanism (MHAM). Such a mechanism has been conceived in order to get, once suitable kinematical constraints are added, what we call an MHAS, i.e., a Modified Hart’s Antiparallelograms Structure. Each of these structures has a stress-free configuration, i.e., a configuration having vanishing deformation energy, which coincides with a circumference. By limiting to the case when all the mechanism bars are rigid and by replacing specific MHAM elements with two bars interconnected by elastic rotational joints, we get a truss structure which, once homogenized into an elastic 1 D continuum, can assume deformed shapes whose deformation energy is not vanishing only when its curvature is not constant. Choosing for the homogenized 1 D continuum, as a deformation measure, the derivative of curvature, and imposing a suitable rescaling of the rotational elastic moduli, we establish the asymptotic micro-macro relationship for its macro-deformation energy.