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A second gradient formulation for a 2D fabric sheet with inextensible fibres

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Abstract

We present numerical simulations of rectangular woven fabrics made of two, initially orthogonal, families of inextensible fibres. We consider an energy functional which includes both first and second gradients of the displacement. The energy density is expressed in terms of the angles between the fibres directions, using trigonometric functions and their gradients. In particular, we focus on an energy density depending on the squared tangent of the shear angle, which automatically satisfies some natural properties of the energy. The numerical results show that final configurations obtained by the second gradient energies are smoother than the first gradient ones. Moreover, we show that if a second gradient energy is considered, the shear energy is better uniformly distributed.

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References

  1. Alibert J., Della Corte A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Z. Angew. Math. Phys. 66(5), 2855–2870 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alibert J.-J., Seppecher P., dell’Isola F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amin Pour, H., Rizzi, N., Salerno, G.: A one-dimensional beam model for single-wall carbon nano tube column buckling. In: Civil-Comp Proceedings (2014)

  4. AminPour H., Rizzi N.: A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis. Math. Mech. Solids 21(2), 168–181 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Andreaus U., Chiaia B., Placidi L.: Soft-impact dynamics of deformable bodies. Contin. Mech. Thermodyn. 25(2-4), 375–398 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Andreaus U., Giorgio I., Lekszycki T.: A 2-d continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time. ZAMM J. Appl. Math. Mech. 94(12), 978–1000 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Assante D., Cesarano C., Fornaro C., Vazquez L.: Higher order and fractional diffusive equations. J. Eng. Sci. Technol. Rev. 8(5), 202–204 (2015) cited By 0

    Google Scholar 

  8. Auffray N., dell’Isola F., Eremeyev V., Madeo A., Rosi G.: Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cao J., Akkerman R., Boisse P., Chen J. et al.: Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results. Compos. Part A Appl. Sci. Manuf. 39(6), 1037–1053 (2008)

    Article  Google Scholar 

  10. Carcaterra, A.: Quantum euler beam—queb: modeling nanobeams vibration. Contin. Mech. Thermodyn. 27(1), 145–156 (2015). doi:10.1007/s00161-014-0341-1

  11. Carcaterra A., dell’Isola F., Esposito R., Pulvirenti M.: Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials. Arch. Ration. Mech. Anal. 218(3), 1239–1262 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cazzani A., Malagù M., Turco E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Contin. Mech. Thermodyn. 28(1-2), 139–156 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. ZAMM J. Appl. Math. Mech. (2016). doi:10.1002/zamm.201500280

  14. Chang C.S., Misra A.: Theoretical and experimental study of regular packings of granules. J. Eng. Mech. 115(4), 704–720 (1989)

    Article  Google Scholar 

  15. Cuomo M., Contrafatto L., Greco L.: A variational model based on isogeometric interpolation for the analysis of cracked bodies. Int. J. Eng. Sci. 80, 173–188 (2014)

    Article  MathSciNet  Google Scholar 

  16. d’Agostino M., Giorgio I., Greco L., Madeo A., Boisse P.: Continuum and discrete models for structures including (quasi-) inextensible elasticae with a view to the design and modeling of composite reinforcements. Int. J. Solids Struct. 59, 1–17 (2015)

    Article  Google Scholar 

  17. Della Corte A., Battista A., dell’Isola F.: Referential description of the evolution of a 2d swarm of robots interacting with the closer neighbors: perspectives of continuum modeling via higher gradient continua. Int. J. Non Linear Mech. 80, 209–220 (2016)

    Article  Google Scholar 

  18. dell’Isola, F., Cuomo, M., Greco, L., Della Corte, A.: Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J. Eng. Math. 1–31 (2016). doi:10.1007/s10665-016-9865-7

  19. dell’Isola F., d’Agostino M., Madeo A., Boisse P., Steigmann D.: Minimization of shear energy in two dimensional continua with two orthogonal families of inextensible fibers: The case of standard bias extension test. J. Elast. 122(2), 131–155 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. dell’Isola, F., Della Corte, A., Giorgio, I., Scerrato, D.: Pantographic 2D sheets: discussion of some numerical investigations and potential applications. Int. J. Non Linear Mech. 80, 200–208 (2016). doi:10.1016/j.ijnonlinmec.2015.10.010

  21. dell’Isola, F., Della Corte, A., Greco, L., Luongo, A.: Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange multipliers and a perturbation solution. Int. J. Solids Struct. 81, 1–12 (2016)

  22. dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. In: Proceedings of Royal Society A, volume 472, page 20150790. The Royal Society (2016)

  23. dell’Isola, F., Seppecher, P., Della Corte, A.: The postulations á la dalembert and á la cauchy for higher gradient continuum theories are equivalent: a review of existing results. In: Proceeding of Royal Society A, volume 471, page 20150415. The Royal Society (2015)

  24. dell’Isola F., Steigmann D., Della Corte A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060804 (2015)

    Article  Google Scholar 

  25. dell’Isola F., Andreaus U., Placidi L.: 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. Math. Mech. Solids 20(8), 887–928 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. dell’Isola F., Lekszycki T., Pawlikowski M., Grygoruk R., Greco L.: Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z. Angew. Math. Phys. 66(6), 3473–3498 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. dell’Isola, F., Placidi, L.: Variational principles are a powerful tool also for formulating field theories. In: dell’Isola, F., Gavrilyuk, S. (eds.) Variational Models and Methods in Solid and Fluid Mechanics, CISM Courses and Lectures, vol. 535, pp. 1–15. Springer, Wien (2011)

  28. dell’Isola F., Steigmann D.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118(1), 113–125 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Di Egidio A., Luongo A., Paolone A.: Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams. Int. J. Non Linear Mech. 42(1), 88–98 (2007)

    Article  MATH  Google Scholar 

  30. Enakoutsa, K., Della Corte, A., Giorgio, I.: A model for elastic flexoelectric materials including strain gradient effects. Math. Mech. Solids 21(2), 242–254 (2016). doi:10.1177/1081286515588638

  31. Gabriele, S., Rizzi, N., Varano, V.: On the imperfection sensitivity of thin-walled frames. In: Civil-Comp Proceedings, vol. 99 (2012)

  32. Gabriele, S., Rizzi, N., Varano, V.: On the postbuckling behaviour of thin walled beams with in-plane deformable cross-sections. In: Civil-Comp Proceedings (2013)

  33. Gabriele, S., Rizzi, N., Varano, V.: A 1d higher gradient model derived from koiter’s shell theory. Math. Mech. Solids, p. 1081286514536721 (2014)

  34. Gabriele, S., Rizzi, N., Varano, V.: A one-dimensional nonlinear thin walled beam model derived from koiter shell theory. In: Civil-Comp Proceedings, vol. 106 (2014)

  35. Gabriele, S., Rizzi, N., Varano, V.: A 1D nonlinear TWB model accounting for in plane cross-section deformation. Int. J. Solids Struct. 94, 170–178 (2016). doi:10.1016/j.ijsolstr.2016.04.017

  36. Giorgio, I., Della Corte, A., dell’Isola, F., Steigmann, D.: Buckling modes in pantographic lattices. Comptes rendus Mecanique. 344(7), 487–501 (2016). doi:10.1016/j.crme.2016.02.009

  37. Greco L., Cuomo M.: An implicit G1 multi patch B-spline interpolation for Kirchhoff–Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Greco L., Cuomo M.: An isogeometric implicit g1 mixed finite element for kirchhoff space rods. Comput. Methods Appl. Mech. Eng. 298, 325–349 (2016)

    Article  MathSciNet  Google Scholar 

  39. Greco, L., Giorgio, I., Battista, A.: In plane shear and bending for first gradient inextesible pantographic sheets: numerical study of deformed shapes and global constraint reactions. Math. Mech. Solids (2016). doi:10.1177/1081286516651324

  40. Grosberg P.: The mechanical properties of woven fabrics part ii: the bending of woven fabrics. Text. Res. J. 36(3), 205–211 (1966)

    Article  Google Scholar 

  41. Grosberg P., Leaf G., Park B.: The mechanical properties of woven fabrics part vi: the elastic shear modulus of plain-weave fabrics. Text. Res. J. 38(11), 1085–1100 (1968)

    Article  Google Scholar 

  42. Grosberg P., Park B.: The mechanical properties of woven fabrics part v: the initial modulus and the frictional restraint in shearing of plain weave fabrics. Text. Res. J. 36(5), 420–431 (1966)

    Article  Google Scholar 

  43. Hearle J.: High-performance fibres. Elsevier, (2001)

  44. Horrocks A., Anand S.: Handbook of technical textiles. Elsevier, (2000)

  45. Luongo A., Zulli D., Piccardo G.: A linear curved-beam model for the analysis of galloping in suspended cables. J. Mech. Mater. Struct. 2(4), 675–694 (2007)

    Article  Google Scholar 

  46. Misra A., Poorsolhjouy P.: Micro-macro scale instability in 2d regular granular assemblies. Contin. Mech. Thermodyn. 27(1-2), 63–82 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Paolone A., Vasta M., Luongo A.: Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces i: non-linear model and stability analysis. Int. J. Non Linear Mech. 41(4), 586–594 (2006)

    Article  Google Scholar 

  48. Pignataro M., Rizzi N., Luongo A.: Stability, bifurcation and postcritical behaviour of elastic structures, volume 39. Elsevier, (2013)

  49. Pignataro M., Rizzi N., Ruta G., Varano V.: The effects of warping constraints on the buckling of thin-walled structures. J. Mech. Mater. Struct. 4(10), 1711–1727 (2010)

    Article  Google Scholar 

  50. Pignataro M., Ruta G., Rizzi N., Varano V.: Effects of warping constraints and lateral restraint on the buckling of thin-walled frames. ASME Int. Mech. Eng. Congr. Expos. 10(B), 803–810 (2010)

    Google Scholar 

  51. Placidi L., Andreaus U., Della Corte A., Lekszycki T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Z. Angew. Math. Phys. 66(6), 3699–3725 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear d4 orthotropic second gradient elastic model. J. Eng. Math. 1–21 (2016). doi:10.1007/s10665-016-9856-8

  53. Potter K.: Bias extension measurements on cross-plied unidirectional prepreg. Compos. A Appl. Sci. Manuf. 33(1), 63–73 (2002)

    Article  Google Scholar 

  54. Rahali Y., Giorgio I., Ganghoffer J.F., Dell’Isola F.: Homogenization à la piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)

    Article  MathSciNet  Google Scholar 

  55. Rinaldi A., Placidi L.: A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices. ZAMM J. Appl. Math. Mech. 94(10), 862–877 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Rivlin, R.S.: Plane strain of a net formed by inextensible cords. J. Rational Mech. Anal. 4(6), 951–974 (1955)

  57. Rizzi N., Varano V.: The effects of warping on the postbuckling behaviour of thin-walled structures. Thin Walled Struct. 49(9), 1091–1097 (2011)

    Article  Google Scholar 

  58. Rizzi N., Varano V.: On the postbuckling analysis of thin-walled frames. In: Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing. Civil-Comp Press (2011)

  59. Rizzi N., Varano V., Gabriele S.: Initial postbuckling behavior of thin-walled frames under mode interaction. Thin Walled Struct. 68, 124–134 (2013)

    Article  Google Scholar 

  60. Ruta G., Pignataro M., Rizzi N.: A direct one-dimensional beam model for the flexural-torsional buckling of thin-walled beams. J. Mech. Mater. Struct. 1(8), 1479–1496 (2006)

    Article  Google Scholar 

  61. Ruta G., Varano V., Pignataro M., Rizzi N.: A beam model for the flexural–torsional buckling of thin-walled members with some applications. Thin Walled Struct. 46(7), 816–822 (2008)

    Article  Google Scholar 

  62. Scerrato D., Giorgio I., Rizzi N.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Z. Angew. Math. Phys. 67(3), 1–19 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  63. Scerrato, D., Zhurba Eremeeva, I., Lekszycki, T., Rizzi, N.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. ZAMM J. Appl. Math. Mech. (2016). doi:10.1002/zamm.201600066

  64. Sciarra G., dell’Isola F., Coussy O.: Second gradient poromechanics. Int. J. Solids Struct. 44(20), 6607–6629 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  65. Scorrano A., Carcaterra A.: Investigation on a nanomechanical transistor. Meccanica 48(8), 1883–1892 (2013)

    Article  MATH  Google Scholar 

  66. Seppecher, P., Alibert, J.-J., Isola, F.D.: Linear elastic trusses leading to continua with exotic mechanical interactions. In: Journal of Physics: Conference Series, vol. 319, p. 012018. IOP Publishing (2011)

  67. Serpieri, R., Della Corte, A., Travascio, F., Rosati, L.: Variational theories of two-phase continuum poroelastic mixtures: a short survey. In: Altenbach, H., Forest, S. (eds.) Generalized Continua as Models for Classical and Advanced Materials, Advanced Structured Materials, vol. 42, pp. 377–394. Springer (2016). ISBN 978-3-319-31719-9

  68. Steigmann D.J., dell’Isola F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mech. Sin. 31(3), 373–382 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  69. Turco E., dell’Isola F., Cazzani A., Rizzi N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z. Angew. Math. Phys. 67(4), 1–28 (2016)

    Article  MathSciNet  Google Scholar 

  70. Turco, E., dell’Isola, F., Rizzi, N.L., Grygoruk, R., Müller, W.N., Liebold, C.: Fiber rupture in sheared planar pantographic sheets: numerical and experimental evidence. Mech. Res. Commun. 76, 86–90 (2016). doi:10.1016/j.mechrescom.2016.07.007

  71. Turco E., Golaszewski M., Cazzani A., Rizzi N.L.: Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete Lagrangian model. Mech. Res. Commun. 76, 51–56 (2016)

    Article  Google Scholar 

  72. Valoroso N., Rosati L.: Consistent derivation of the constitutive algorithm for plane stress isotropic plasticity. part i: theoretical formulation. Int. J. Solids Struct. 46(1), 74–91 (2009)

    Article  MATH  Google Scholar 

  73. Wang C.M., Zhang Hui, Gao R.P., Duan W.H., Challamel N.: Hencky bar-chain model for buckling and vibration of beams with elastic end restraints. Int. J. Struct. Stab. Dyn. 15(07), 1540007 (2015)

    Article  MathSciNet  Google Scholar 

  74. Yang Y., Ching W.Y., Misra A.: Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film. J. Nanomech. Micromech. 1(2), 60–71 (2011)

    Article  Google Scholar 

  75. Yang Y., Misra A.: Higher-order stress-strain theory for damage modeling implemented in an element-free galerkin formulation. Comput. Model. Eng. Sci.: CMES 64(1), 1–36 (2010)

    MathSciNet  MATH  Google Scholar 

  76. Yang Y., Misra A.: Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. Int. J. Solids Struct. 49(18), 2500–2514 (2012)

    Article  Google Scholar 

  77. Zhu B., Yu T.X., Tao X.M.: Large deformation and slippage mechanism of plain woven composite in bias extension. Compos. Part A Appl. Sci. Manuf. 38(8), 1821–1828 (2007)

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Engineering, International Telematic University Uninettuno, Rome, Italy

    Luca Placidi

  2. University of Catania, Catania, Italy

    Leopoldo Greco

  3. Otto-Von-Guericke-University Magdeburg, Magdeburg, Germany

    Sara Bucci

  4. University of Sassari, Alghero, Italy

    Emilio Turco

  5. Roma Tre University, Rome, Italy

    Nicola Luigi Rizzi

  6. MEMOCS, International Research Center on Mathematics and Mechanics of Complex Systems, Cisterna di Latina, Italy

    Luca Placidi, Leopoldo Greco, Sara Bucci, Emilio Turco & Nicola Luigi Rizzi

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  1. Luca Placidi
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  2. Leopoldo Greco
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  3. Sara Bucci
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  4. Emilio Turco
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  5. Nicola Luigi Rizzi
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Correspondence to Luca Placidi.

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Placidi, L., Greco, L., Bucci, S. et al. A second gradient formulation for a 2D fabric sheet with inextensible fibres. Z. Angew. Math. Phys. 67, 114 (2016). https://doi.org/10.1007/s00033-016-0701-8

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