Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model

By Luca Placidi, Ugo Andreaus, Ivan Giorgio. 2017

182 citations recorded by the source corpus.

A linear elastic second gradient orthotropic two-dimensional solid that is invariant under $$90^{\circ }$$90∘ rotation and for mirror transformation is considered. Such anisotropy is the most general for pantographic structures that are composed of two identical orthogonal families of fibers. It is well known in the literature that the corresponding strain energy depends on nine constitutive parameters: three parameters related to the first gradient part of the strain energy and six parameters related to the second gradient part of the strain energy. In this paper, analytical solutions for simple problems, which are here referred to the heavy sheet, to the nonconventional bending, and to the trapezoidal cases, are developed and presented. On the basis of such analytical solutions, gedanken experiments were developed in such a way that the whole set of the nine constitutive parameters is completely characterized in terms of the materials that the fibers are made of (i.e., of the Young’s modulus of the fiber materials), of their cross sections (i.e., of the area and of the moment of inertia of the fiber cross sections), and of the distance between the nearest pivots. On the basis of these considerations, a remarkable form of the strain energy is derived in terms of the displacement fields that closely resembles the strain energy of simple Euler beams. Numerical simulations confirm the validity of the presented results. Classic bone-shaped deformations are derived in standard bias numerical tests and the presence of a floppy mode is also made numerically evident in the present continuum model. Finally, we also show that the largeness of the boundary layer depends on the moment of inertia of the fibers.

Subfields: Materials Chemistry, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/346595750bdd8b85c15bc71d6712b891b2e455a2

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In Mechanics of Materials

Citations in the corpus

Linear pantographic sheets: Asymptotic micro-macro models identification (2017, background): “This paper, differently to what done in [83], considers the case of extensible fibers and gives a more solid foundation to and generalizes the heuristic results presented in [82], based on an accurate analysis of the different and relative order of magnitude of involved physical phenomena.”
Strain gradient and generalized continua obtained by homogenizing frame lattices (2018, methodology): “Structures based on pantographic mechanisms have been intensively studied from theoretical [Placidi et al. 2016] but also numerical [Turco et al. 2017; Harrison 2016] and experimental [Placidi et al. 2017] points of view.”
Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models (2016, background): “The mechanical systems which could be described by the introduced discrete model include those considered and studied in [2,4,5,10,11].”
Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies (2016, background): “For a detailed discussion of the theoretical prevision of the thickness of the boundary layer in this type of structure, see [66].”
Exegesis of the Introduction and Sect. I from “Fundamentals of the Mechanics of Continua”** by E. Hellinger (2017, background): “Again the ideas presented by Hellinger have been continued after many years of apparent neglect: among those papers which seems to us have been animated by Hellinger’s spirit we found particularly interesting: [5, 7, 10, 27, 39, 59, 73–75, 83, 90] for the generalized theory of beams and [6, 8, 13, 1”
Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions (2017, background): “In [50, 51] the integration by part procedure valid for boundaries with corner points is considered.”