From articulated-bi-parallelograms to third gradient 1D continua

Recently, it has been conjectured that a specific modular articulated bi-parallelogram microstructure is a solution for the problem of synthesis for a 1D continuum whose deformation elastic energy depends on the curvature gradient (dell’Isola et al. [ Math. Mech. Complex Syst. 12 (2024)] and Terranova et al. [ Comptes Rendus. Mécanique 353 (2025)] ). In this paper, we present an asymptotic procedure for getting the homogenized description of that microstructured system under large in-plane deformation. It is shown that such a periodic structure behaves macroscopically as a third gradient 1D continuum. The elastic energy stored in a module of such a microstructured system deformed by a gradient of curvature combined with extension is first established. This leads to the energy density of the effective 1D continuum. The latter involves three elastic stiffness coefficients related to the curvature gradient, the elongation, and the coupling of both, whose expressions are explicitly related to the morphology of the module, the stiffnesses of the micro bars, and the curvature. The strong formulation of the equilibrium condition of the microstructured system is then deduced following the Euler–Lagrange method of minimization of energy. The calculation of the first variation of the deformation energy allows for the determination of the generalized external forces which can be applied to the equivalent 1D continuum whose deformation energy depends on the gradient of curvature and extension: that is, normal and transverse force together with couple and double couple. Consequently, we determine the corresponding balance equations and the constitutive equations for forces and both couple and double couples. Some numerical examples are given in the case of newly introduced 1D continua loaded at their extremity by couples and double couples.