1D continua with internal variable as homogenized models for modified Hart’s antiparallelograms micro-structures

In this paper, we show, via an asymptotic expansion, that a 1D continuum model, with an internal variable, is suitable to describe the mechanical behavior of the Inverted Hart’s micro-architectures, generalizing the treatment presented in Rizzi et al.(2026). An internal variable is needed to describe the kinematics of the 1D continuum: we find an expression for deformation energy, which depends on the gradient of curvature of a 1D continuum and on the introduced internal variable. We find, as a particular case, the results already presented in Rizzi et al.(2026): when the condensation of kinematical parameters is possible, the energy density of the 1D model depends only on the curvature gradient.