A Review on Models for the 3D Statics and 2D Dynamics of Pantographic Fabrics
A review on models for the statics of out-of-plane deformable pantographic fabrics is presented, along with a model describing the dynamics of in-plane-only deformable pantographic fabrics. We discuss those models able to describe the mechanical exotic pr
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Abstract A review on models for the statics of out-of-plane deformable pantographic fabrics is presented, along with a model describing the dynamics of in-planeonly deformable pantographic fabrics. We discuss those models able to describe the mechanical exotic properties conferred by the peculiar microstructure possessed by pantographic metamaterials, when three-dimensional deformations and in-plane dynamics are separately involved. For each approach, model formulation and modelling assumptions are discussed along with the presentation of numerical solutions in exemplary cases, and no attempt is made to model damage and failure phenomena. Keywords Pantographic fabrics ⋅ Metamaterials ⋅ Higher gradient continua Review ⋅ Microstructure ⋅ Dynamics ⋅ Out-of-plane deformations
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1 Introduction In the wide current research literature about mechanical and acoustic metamaterials, pantographic fabrics [1, 9, 16, 18, 20, 22, 23, 26, 28, 29, 37, 39, 40, 58, 67, 69, 71] (see Fig. 1) represent one of the very promising topics, being both of great practical and theoretical interest. Indeed, their potentially high strength-to-weight ratio yields pantographic fabrics particularly suitable for some class of applications (see [4, 36, 38]), while their great theoretical interest derives from the fact that, in order to describe their exotic phenomenology, one has to resort to higher gradient continuum theories [5, 45] or micromophic theories [2, 49] with the related problem of homogenization [59] and of different strategies for the numerical integration E. Barchiesi Department of Structural and Geotechnical Engineering, Universitá di Roma La Sapienza, Rome, Italy L. Placidi (✉) International Telematic University Uninettuno, C.so Vittorio Emanuele II, 39 00186 Rome, Italy e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2017 M.A. Sumbatyan (ed.), Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, Advanced Structured Materials 59, DOI 10.1007/978-981-10-3797-9_14
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E. Barchiesi and L. Placidi
Fig. 1 A pantographic structure
[8, 10]. The field of fiber-reinforced materials [31, 44, 53, 54, 68] and of elastic sheets with bending stiffness [41–43, 55, 60] is certainly not new in the literature. In the following sections we are going to present a concise survey of mathematical models which have been purposely developed for out-of-plane deformable pantographic structures, with regard to their statics, and for in-plane deformable ones, with regard to their dynamics. In [57] a review on models for the description of the mechanical behavior, at equilibrium, of pantographic sheets undergoing planar deformations only, i.e. models whose placement functions are 𝜒 ∶ ℝ2 ⊇ B → ℝ2 , is presented. The present paper can hence be considered a sequel of [57] as, instead of considering only placement functions 𝜒 ∶ ℝ2 ⊇ B → ℝ2 , we consider herein also placement functions 𝜒 ∶ ℝ2 ⊇ B → ℝ3 and 𝜒 ∶ ℝ3 ⊇ B → ℝ3 , thus allowing outof-plane deformations. Moreover, w
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