Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models

Abstract

Hencky (U ̈ber die angen ̈aherte L ̈osung von Stabilit ̈atsproblemen im Raum mittels der elastischen Gelenkkette. Ph.D. thesis, Engelmann, 1921) proposed a discrete model for elasticae by introducing rigid bars and rotational springs. Hencky (Proc R Soc Lond A Math Phys Eng Sci 472(2185), 2016) approach has been introduced to heuristically motivate the need of second gradient continua. Here, we present a novel numerical code implementing directly the discrete Hencky- type model which is robust enough to solve the problem of the determination of equilibrium configurations in the large deformation and displacement regimes. We apply this model to study some potentially applicable problems, and we compare its performances with those of the second gradient continuum model. The numerical evidence presented supports the conjecture that Hencky-type converges to second gradient model

Towards comprehensive characterisation and modelling of the forming and wrinkling mechanics of engineering fabrics

“…In this investigation it will also be employed to identify both the in-plane bending stiffness (D'Agostino et al, 2015;Dell'Isola and Steigmann, 2014;Ferretti et al, 2014;Giorgio et al, 2016;Harrison, 2016;Scerrato et al, 2016;Turco et al, 2016) and the torsional stiffness of the sheared fabric (Lomov and Verpoest, 2006;Steigmann and Dell'Isola, 2015) by monitoring the sample kinematics, including the shear angle at the centre of the specimen (D'Agostino et al, 2015;Ferretti et al, 2014;Harrison, 2016) and the out-of-plane wrinkling behaviour (Arnold et al, 2016;Boisse et al, 2011;Cherouat and Billoët, 2001;Dangora et al, 2015;Harrison, 2016;ten Thije and Akkerman, 2009;Thompson et al, 2016).…”

“…• the tensile stiffness in each of the two fibre directions (Boisse et al, 2001;Potluri and Thammandra, 2007) • the fabric shear stiffness (resistance to trellis shear) (Boisse et al, 2016;Cao et al, 2008;Harrison et al, 2012Harrison et al, , 2008 • the out-of-plane bending stiffness in each of the two fibre directions (Cooper, 1960;de Bilbao et al, 2010;Harrison, 2016;Hu, 2004;ISO, 1998;Lammens et al, 2014;Lomov et al, 2003;Peirce, 1930;Plaut, 2015) • the in-plane bending stiffness in each of the two fibre directions (D'Agostino et al, 2015;Dell'Isola and Steigmann, 2014;Ferretti et al, 2014;Giorgio, 2016;Giorgio et al, 2016;Harrison, 2016;Scerrato et al, 2016;Steigmann and Dell'Isola, 2015;Turco et al, 2016) • the torsional stiffness in each of the two fibre directions (Cooper, 1960;D'Agostino et al, 2015;Giorgio, 2016;Giorgio et al, 2016;Lomov and Verpoest, 2006;Steigmann and Dell'Isola, 2015) • inter-ply and tool-ply friction (Sachs et al, 2014) • the thickness of the sheet (Chen and Ye, 2006;Pazmino et al, 2014) In this investigation, experimental identification of the shear stiffness, out-of-plane bending, in-plane bending and torsional stiffness is demonstrated using just two simple tests; a cantilever bending test (Cooper, 1960;Harrison, 2016;…”

Fiber rupture in sheared planar pantographic sheets: Numerical and experimental evidence

“…As discussed in full detail in [3] Lagrangian discrete models can be a viable option in the description of pantographic sheets: the possibility to predict their shapes in large deformation equilibrium configurations using said discrete models has been the object of the investigations presented in [4]. Referring the reader to just cited papers for a detailed discussion of the model and the algorithmic procedure we use to determine equilibrium configurations, we delineate here the main ideas on which we base our solution strategy when dealing with both elastic and rupture deformation phenomena.…”

“…The algorithm which is successfully used here (following the ideas proposed in this context in [3]) handles the strong nonlinearities arising in the aforementioned minimization process by proceeding to calculate sequences of close equilibrium configurations relative to close boundary conditions. The relevant economy of calculation time is exploited also in the simulations we present here for elastic regimes.…”

Reconfiguration of a smart surface using heteroclinic connections

“…However, the work in this paper analyses the vertical deformation and so is quite different to the Hencky-type discrete model, which focuses on planar deformation [33]. In §2, the surface structure is considered as an elastic plane which has a range of both stable and unstable configurations.…”