Stepwise analysis of pantographic beams subjected to impulsive loads

Abstract

Materials based on pantographic unit cells have very interesting mechanical peculiarities. For these reasons they are largely studied from a theoretical, experimental, and numerical point of view. Numerical simulations furnish an important contribution for the the design and optimization of such materials and, more generally, for metamaterials. Here, we consider the influence of inertial forces, removing the hypothesis of quasistatic loading. By using an intrinsically discrete model, inspired by Hencky’s ideas, already tested in a series of published works, here we add the contribution of inertial forces and, in the framework of stepwise schemes, we re-experience an adaptive integration scheme capable of reconstructing the best structural response corresponding to a prefixed time step. Several numerical simulations, although preparatory, inspire some remarks on materials based on pantographic cells and outline the way for future challenges.

Modeling of three-dimensional beam nonlinear vibrations generalizing Hencky’s ideas

“…Probably, the lack of details is the main reason of the little success of the method. A complete description of Casciaro’s scheme along with correct formulas for optimal values of the parameters α and β is reported in Turco [24].…”

“…having used the positions: q 2 = q Á q and E q u = q × u for any vector u. This version of the Rodrigues' formula is perfectly equivalent to (24), but allows to avoid some singularities in the algebraic manipulations necessary to define quantities such as the strain energy and its first and second derivatives which are needed by the numerical algorithm employed in the sequel to reconstruct the equilibrium path of the structure. In addition, the use of q defined by equation ( 26) makes the rotation vector vary in the same range, from À' to + ', of the displacements vector and therefore, in principle, is beneficial for the wellconditioning of the equilibrium equations to be solved.…”

“…In rough terms, the key idea of Casciaro's numerical integration scheme consists in the introduction of two free parameters which will be used for optimizing the performance of the numerical integration scheme on the basis of the chosen time step length. A detailed description of the Casciaro's integration scheme is reported in the recent review work [24].…”

A numerical investigation on impulse-induced nonlinear longitudinal waves in pantographic beams

“…Actually, a more detailed description of the integration scheme can be found in [61], although both [60] and [61] contain some typos in the formulas. Recently, a more thorough analysis of the Casciaro's scheme has been carried out in [28], which presents some numerical examples and analytical computations providing insights into the method, specifically into the rationale leading to the optimal expressions (13) and (14).…”

“…The largest part by far of research on pantographic materials has been focused on the analysis of small and large deformation static regimes. On the other hand, contributions dealing with the dynamical response of pantographic materials are rare in the scientific literature, some examples being [27][28][29]. The development of reliable modeling and computational tools addressing the dynamics of pantographic materials could accelerate the exploitation of such materials in cases of practical interest, where the influence of inertia forces cannot be neglected.…”