Computational identification of double-bending stiffness: from Zigzagged Articulated Parallelograms with Articulated Braces (ZAPAB) structures to pure-curvature gradient planar inextensible 1D continua

An inextensible 1D continuum whose deformation energy purely depends on the gradient of the associated curvature is introduced to describe the behavior of Zigzagged Articulated Parallelograms with Articulated Braces truss structures (ZAPAB structures) after homogenization. We choose a particular ZAPAB structure in which all but one of the constituting bars of the basic module do not change their length under applied loads. This judicious choice allows us to verify, through numerical simulations, that the corresponding 1D continuum indeed has a deformation energy that depends solely on the derivative of curvature. Thus, by employing a best-fitting approach based on the least squares method, we numerically identify the best stiffness coefficient (in the least squared sense) associated with the energy contribution due to the gradient of curvature, termed as double-bending stiffness. The presented simulations consider the case of uniformly distributed applied dead loads, and reveal a strong match between the current configurations of the proposed 1D continuum model, obtained numerically through the Finite Element Method, and the current configurations of the ZAPAB structure (for a selected number of basic modules), obtained through a discrete numerical approach, with the curves coinciding up to certain intrinsic error. These results require the development of an analytical micro–macro identification procedure. ZAPAB structures facilitate advances in the synthesis of tailored materials and the n -th gradient theory. We adopt a theory-driven approach with the expectation of devising materials with exotic behaviors. Specifically, we anticipate that material lines capable of not storing deformation energy under uniform bending (constant curvature) will be obtained after homogenization, thereby paving the way for future work that introduces complex materials built upon them. Our discussion is inspired by well-known pantographic structures, which serve as archetypes of second gradient materials designed in such a way that no deformation energy is stored under uniform extension.