Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycl
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O R I G I NA L A RT I C L E
Ivan Giorgio
Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycloidal metamaterial
Received: 6 October 2020 / Accepted: 9 November 2020 © The Author(s) 2020
Abstract A nonlinear elastic model for nets made up of two families of curved fibers is proposed. The net is planar prior to the deformation, but the equilibrium configuration that minimizes the total potential energy can be a surface in the three-dimensional space. This elastic surface accounts for the stretching, bending, and torsion of the constituent fibers regarded as a continuous distribution of Kirchhoff rods. A specific example of fiber arrangement, namely a cycloidal orthogonal pattern, is examined to illustrate the predictive abilities of the model and assess the limit of applicability of it. A numerical micro–macro-identification is performed with a model adopting a standard continuum deformable body at the level of scale of the fibers. A few finite element simulations are carried out for comparison purposes in statics and dynamics, performing modal analysis. Finally, a topology optimization problem has been carried out to change the macroscopic shear stiffness to enlarge the elastic regime and reduce the risk of damage without excessively losing bearing capacity. Keywords Nonlinear elasticity · Second gradient models · Homogenized nets · Metamaterials 1 Introduction The advent of new manufactory technologies [22,45,56], like 3D printing, and the increasing demand for materials with high-performance render desirable design techniques for advanced forms of micro-structured materials have been but little considered heretofore. Accordingly, many research lines have been initiated with the aim of setting up general methods of synthesis of micro-structures for obtaining materials with a macroscopic behavior designed for satisfying given requirements [9,12,14,57,73,79,80,82]. These materials conceived for fulfilling one or more given functions belong to the set of the generalized continuum materials for which is their microstructure that confers the desired properties. This point of view is what characterizes the so-called metamaterials. Although their first appearance was in the field of electromagnetic and optical applications, soon enough there have been developments also in the mechanical field (see, e.g., [5,15,20,48, 69]). The rationale for the synthesis of such materials is quite ancient. Indeed, the same approach can be found in the synthesis of mechanisms [55,62,64,66] or electric networks [11]. In both cases, there are elemental components with specific properties that, if properly connected to each other, are capable of producing a whole system providing the desired functioning. However, since the introduction of metamaterials is rather recent, a systematic way to design micro-structures is not yet fully developed, as in the case of mechanisms and electric networks. Therefore, the metamaterials proposed up to now in the literature are the results
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