Discrete versus homogenized continuum modeling in finite deformation bias extension test of bi-pantographic fabrics
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O R I G I NA L A RT I C L E
E. Barchiesi
· J. Harsch · G. Ganzosch · S. R. Eugster
Discrete versus homogenized continuum modeling in finite deformation bias extension test of bi-pantographic fabrics
Received: 29 March 2020 / Accepted: 14 August 2020 © The Author(s) 2020
Abstract A 2D-continuum model describing finite deformations in plane of discrete bi-pantographic fabrics has been recently obtained by applying an asymptotic procedure based on a set of local generalized coordinates. Rectangular bi-pantographic prototypes were additively manufactured by selective laser sintering using polyamide as raw material. Displacement-controlled bias extension tests were performed on such specimens for total elastic deformations up to ca. 25%. Experimental force measurements, complemented by discrete displacement measurements obtained by local digital image correlation, were used to fit the continuum model. In the present paper, a global and minimal set of generalized coordinates, alternative to the one used for the homogenization, is introduced for the discrete model. The mechanical constitutive parameters appearing in the discrete model are then found by means of collected experimental data. Finally, a comparison between experiments, the discrete and the continuum model is presented. It is concluded that (a) the discrete model and the experimental data are in excellent agreement, and that (b) the continuum retains the relevant phenomenology of the discrete system even for a rather low number of cells. Keywords Bi-pantographic fabrics · Second gradient continua · Discrete spring models · Additive manufacturing · Experimental mechanics
1 Introduction Recently, [1,2] presented the derivation by an asymptotic homogenization procedure [3–6] of a 1D-continuum model [7–11] being capable of describing finite deformations in plane of a discrete spring [12–19] pantographic structure [20–27] looking like an expanding barrier, referred to as pantographic beam. Based on such results, Barchiesi et al. [28,29] generalized to finite deformations the homogenization of bipantographic fabrics, first achieved by Seppecher et al. [6]. Bi-pantographic fabrics are conceived as assemblies Communicated by Luca Placidi. E. Barchiesi (B) Dipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma “La Sapienza”, Rome, Italy E-mail: [email protected] E. Barchiesi · S. R. Eugster International Research Center M&MoCS, Università degli Studi dell’Aquila, L’Aquila, Italy J. Harsch · S. R. Eugster Institute for Nonlinear Mechanics, University of Stuttgart, Stuttgart, Germany G. Ganzosch Institut für Mechanik, TU Berlin, Berlin, Germany
E. Barchiesi et al.
Fig. 1 Additively manufactured bi-pantographic rectangular specimen
of discrete pantographic beams (see Fig. 1) leading at macroscopic scale to second gradient materials in plane. (For a representative account of second gradient and generalized continua, the reader is referred to [30–39].) For such materials, the deformation energy density depends upon the second g
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