In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions

The aim of the present paper is the analysis of a two-dimensional continuum with two families of inextensible fibers that are orthogonal in the initial configuration. In the first part of the work, a new formulation is presented, in which the problem is reduced to a standard nonlinear constrained minimization, while in the second part of the work several numerical investigations are presented considering different boundary conditions with respect to standard symmetric bias extensional tests. The conceptual framework can be recognized in the researches by Pipkin and Rivlin on inextensible nets. Furthermore, an implicit version of the Rivlin representation of the generic placement for a two-dimensional sheet with two families of inextensible fibers is provided by considering the angles of the fiber directors as degrees of the freedom of the formulation. In this way the first gradient formulation is given in terms of two angle fields only.

[1]  Giuseppe Piccardo,et al.  A complete dynamic approach to the Generalized Beam Theory cross-section analysis including extension and shear modes , 2014 .

[2]  I. Verpoest,et al.  Model of shear of woven fabric and parametric description of shear resistance of glass woven reinforcements , 2006 .

[4]  Nicola Luigi Rizzi,et al.  The effects of warping on the postbuckling behaviour of thin-walled structures , 2011 .

[5]  Francesco dell’Isola,et al.  Buckling modes in pantographic lattices , 2016 .

[6]  Ugo Andreaus,et al.  Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle , 2002 .

[7]  A.P.S. Selvadurai,et al.  Torsion of a layered composite strip , 2013 .

[8]  N. Roveri,et al.  Damage detection in structures under traveling loads by Hilbert–Huang transform , 2012 .

[9]  Andrew C. Long,et al.  Shear characterisation of viscous woven textile composites: a comparison between picture frame and bias extension experiments , 2004 .

[10]  A. Milani,et al.  A comparative analysis of a modified picture frame test for characterization of woven fabrics , 2009 .

[11]  S. Ebeid,et al.  Comparison of picture frame and Bias-Extension tests for the characterization of shear behaviour in natural fibre woven fabrics , 2013, Fibers and Polymers.

[12]  Ignace Verpoest,et al.  Carbon composites based on multiaxial multiply stitched preforms. Part 3: Biaxial tension, picture frame and compression tests of the preforms , 2005 .

[13]  Nicola Rizzi,et al.  The effects of warping constraints on the buckling of thin-walled structures , 2010 .

[14]  R. Postle,et al.  Experimental Methods for Measuring Fabric Mechanical Properties: A Review and Analysis , 1999 .

[15]  M. Cuomo,et al.  Simplified analysis of a generalized bias test for fabrics with two families of inextensible fibres , 2016 .

[16]  Giuseppe Rosi,et al.  Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces , 2015 .

[17]  Francesco dell’Isola,et al.  Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence , 2015 .

[18]  Francesco dell’Isola,et al.  Plane bias extension test for a continuum with two inextensible families of fibers: A variational treatment with Lagrange multipliers and a perturbation solution , 2016 .

[19]  J. Ganghoffer,et al.  A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure , 2013, Biomechanics and Modeling in Mechanobiology.

[20]  Giovanni Solari,et al.  Serviceability criteria for wind-induced acceleration and damping uncertainties , 1998 .

[21]  Giuseppe Piccardo,et al.  Non-linear discrete models for the stochastic analysis of cables in turbulent wind , 2010 .

[22]  David J. Steigmann,et al.  A model for frictional slip in woven fabrics , 2003 .

[23]  A. Misra,et al.  Micromechanical model for viscoelastic materials undergoing damage , 2013 .

[24]  Giuseppe Piccardo,et al.  A GBT Model for the Analysis of Composite Steel–Concrete Beams with Partial Shear Interaction , 2015 .

[25]  Yang Yang,et al.  Higher-Order Continuum Theory Applied to Fracture Simulation of Nanoscale Intergranular Glassy Film , 2011 .

[26]  Ignacio Carol,et al.  Crack opening conditions at ‘corner nodes’ in FE analysis with cracking along mesh lines , 2007 .

[27]  Gilles Hivet,et al.  Experimental analysis of the influence of tensions on in plane shear behaviour of woven composite reinforcements , 2008 .

[28]  Giuseppe Ruta,et al.  A beam model for the flexural–torsional buckling of thin-walled members with some applications , 2008 .

[29]  Paul Steinmann,et al.  Isogeometric analysis of 2D gradient elasticity , 2011 .

[30]  Stefano Gabriele,et al.  Initial postbuckling behavior of thin-walled frames under mode interaction , 2013 .

[31]  A. Pipkin,et al.  Some developments in the theory of inextensible networks , 1980 .

[32]  Antonio Cazzani,et al.  Isogeometric analysis of plane-curved beams , 2016 .

[33]  Angelo Luongo,et al.  Controlling the Limit-Cycle of the Ziegler Column via a Tuned Piezoelectric Damper , 2015 .

[34]  Tomasz Lekszycki,et al.  Modeling of an initial stage of bone fracture healing , 2015 .

[35]  A.P.S. Selvadurai,et al.  Concentrated loading of a fibre-reinforced composite plate: Experimental and computational modeling of boundary fixity , 2014 .

[36]  Luca Placidi,et al.  A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model , 2016 .

[37]  Jean-François Ganghoffer,et al.  Equivalent mechanical properties of textile monolayers from discrete asymptotic homogenization , 2013 .

[38]  Dionisio Del Vescovo,et al.  Theoretical and experimental dynamic analysis aimed at the improvement of an acoustic method for fresco detachment diagnosis , 2009 .

[39]  Victor A. Eremeyev,et al.  Material symmetry group of the non-linear polar-elastic continuum , 2012 .

[40]  Flavio Stochino,et al.  Constitutive models for strongly curved beams in the frame of isogeometric analysis , 2016 .

[41]  Andrew C. Long,et al.  Normalization of Shear Test Data for Rate-independent Compressible Fabrics , 2008 .

[42]  Cung Huy Nguyen,et al.  Aeroelastic instability and wind-excited response of complex lighting poles and antenna masts , 2015 .

[43]  Pierre Seppecher,et al.  A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium , 1997 .

[44]  A. Della Corte,et al.  The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[45]  Francesco dell’Isola,et al.  A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio‐resorbable materials , 2012 .

[46]  Victor A. Eremeyev,et al.  Material symmetry group and constitutive equations of micropolar anisotropic elastic solids , 2016 .

[47]  A. Pipkin,et al.  PLANE TRACTION PROBLEMS FOR INEXTENSIBLE NETWORKS , 1981 .

[48]  Gabriel Wittum,et al.  Remodelling in statistically oriented fibre-reinforced materials and biological tissues , 2015 .

[49]  Francesco dell’Isola,et al.  Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies , 2017 .

[50]  Ugo Andreaus,et al.  Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation , 2016 .

[51]  Giuseppe Rosi,et al.  On the failure of the ‘Similar Piezoelectric Control’ in preventing loss of stability by nonconservative positional forces , 2015 .

[52]  Alfio Grillo,et al.  Elasticity and permeability of porous fibre-reinforced materials under large deformations , 2012 .

[53]  Gilles Hivet,et al.  A contribution to the analysis of the intrinsic shear behavior of fabrics , 2011 .

[54]  Francesco dell’Isola,et al.  Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching , 2015 .

[55]  Ugo Andreaus,et al.  Soft-impact dynamics of deformable bodies , 2013 .

[56]  M. Fagone,et al.  Experimental global analysis of the efficiency of carbon fiber anchors applied over CFRP strengthened bricks , 2014 .

[57]  Ugo Andreaus,et al.  Modeling of Trabecular Architecture as Result of an Optimal Control Procedure , 2013 .

[58]  Claude Boutin,et al.  Generalized inner bending continua for linear fiber reinforced materials , 2011 .

[59]  Luisa Pagnini,et al.  Reliability analysis of wind-excited structures , 2010 .

[60]  Leopoldo Greco,et al.  An isogeometric implicit G1 mixed finite element for Kirchhoff space rods , 2016 .

[61]  G. Rosi,et al.  Piezoelectric control of Hopf bifurcations: A non-linear discrete case study , 2016 .

[62]  Alessandro Della Corte,et al.  Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof , 2015 .

[63]  Wojciech Pietraszkiewicz,et al.  The Nonlinear Theory of Elastic Shells with Phase Transitions , 2004 .

[64]  Francesco dell’Isola,et al.  Synthesis of Fibrous Complex Structures: Designing Microstructure to Deliver Targeted Macroscale Response , 2015 .

[65]  R. Rivlin.  Plane Strain of a Net Formed by Inextensible Cords , 1955 .

[66]  Tomasz Lekszycki,et al.  Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients , 2015 .

[67]  Francesco dell’Isola,et al.  A Two-Dimensional Gradient-Elasticity Theory for Woven Fabrics , 2015 .

[68]  F. dell’Isola,et al.  Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium , 2016, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[69]  Characterisation of material properties for draping of dry woven composite material , 2003 .

[70]  Ugo Andreaus,et al.  At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola , 2013, 1310.5599.

[71]  M. Pulvirenti,et al.  Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials , 2015, 1504.08015.

[72]  Luca Placidi,et al.  A variational approach for a nonlinear 1-dimensional second gradient continuum damage model , 2015 .

[73]  P. Potluri,et al.  Characterising the shear–tension coupling and wrinkling behaviour of woven engineering fabrics , 2012 .

[74]  Giuseppe Piccardo,et al.  A direct approach for the evaluation of the conventional modes within the GBT formulation , 2014 .

[75]  P. Harrison,et al.  Evaluation of normalisation methods for uniaxial bias extension tests on engineering fabrics , 2014 .