Advertisement

Quantitative analysis of deformation mechanisms in pantographic substructures: experiments and modeling

  • Emilio Turco
  • Anil Misra
  • Rizacan Sarikaya
  • Tomasz Lekszycki
Original Article

Abstract

In order to get detailed information about the mechanical behavior of pantographic elementary substructure and elements, small-scale specimens were sintered using polyamide powder, constituted by three orthogonal pairs of beams interconnected through pivots forming pantographic cells. The mechanical properties of interconnecting pivots and constituting beams are investigated by comparing experimental evidence with an enhanced Piola–Hencky model. The careful agreement between experimental and predicted results allows us to estimate: (i) the macro-shear stiffness of interconnecting pivots (corresponding to micro-torsional stiffness), (ii) extensional stiffness and (iii) bending stiffness of constituting beams.

Keywords

Pantographic sheets Discrete models Nonlinear analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    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. Zeitschrift für angewandte Mathematik und Physik 66(6), 3473–3498 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    dell’Isola, F., Della Corte, A., Greco, L., Luongo, A., 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)CrossRefGoogle Scholar
  3. 3.
    dell’Isola, F., Cuomo, M., Greco, L., Della, A.: Corte. Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J. Eng. Math. (2016).  https://doi.org/10.1007/s10665-016-9865-7 zbMATHGoogle Scholar
  4. 4.
    Steigmann, D.J., Pipkin, A.C.: Equilibrium of elastic nets. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 335(1639), 419–454 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    dell’Isola, F., Steigmann, D.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118(1), 113–125 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)CrossRefGoogle Scholar
  7. 7.
    Harrison, P.: Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh. Compos. Part A Appl. Sci. Manuf. 81, 145–157 (2016)CrossRefGoogle Scholar
  8. 8.
    dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472(2185), 20150790 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A Math. Phys. Eng. Sci. 473, 20170636 (2017)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Alibert, J.-J., Della, A.: Corte. Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Zeitschrift für Angewandte Mathematik und Physik 66(5), 2855–2870 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9(5), 241–257 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Javili, A., dell’Isola, F., Steinmann, P.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids 61(12), 2381–2401 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bertram, A.: Finite gradient elasticity and plasticity: a constitutive finite gradient elasticity and plasticity: a constitutive thermodynamical framework. Contin. Mech. Thermodyn. 28, 869–883 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Auffray, N., dell’Isola, F., Eremeyev, V.A., Madeo, A., Rosi, G.: Analytical continuum mechanics á la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Greco, L., Giorgio, I., Battista, A.: In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions. Math. Mech. Solids 22(10), 1950–1975 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Eremeyev, V.A., dell’Isola, F., Boutin, C., Steigmann, D.: Linear pantographic sheets: existence and uniqueness of weak solutions. J. Elast. 2017, 1–22 (2017)Google Scholar
  20. 20.
    Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math. Mech. Solids 21(2), 210–221 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bertram, A., Glüge, R.: Gradient materials with internal constraints. Math. Mech. Complex Syst. 4(1), 1–15 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Milton, G., Briane, M., Harutyunyan, D.: On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials. Math. Mech. Complex Syst. 5(1), 41–94 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Chatzigeorgiou, G., Javili, A., Steinmann, P.: Multiscale modelling for composites with energetic interfaces at the micro- or nanoscale. Math. Mech.Solids 20(9), 1130–1145 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Contin. Mech. Thermodyn. 28(1), 215–234 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Misra, A., Poorsolhjouy, P.: Grain- and macro-scale kinematics for granular micromechanics based small deformation micromorphic continuum model. Mech. Res. Commun. 81, 1–6 (2017)CrossRefGoogle Scholar
  26. 26.
    Khakalo, S., Niiranen, J.: Form II of Mindlin’s second strain gradient theory of elasticity with a simplification: for materials and structures from nano- to macro-scales. Eur. J. Mech. A/Solids (to appear), (2018)Google Scholar
  27. 27.
    Khakalo, S., Balobanov, V., Niiranen, J.: Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics. J. Eng. Sci. 127, 33–52 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für Angewandte Mathematik und Physik 67(4), 1–28 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Turco, E., Giorgio, I., Misra, A., dell’Isola, F.: King post truss as a motif for internal structure of (meta)material with controlled properties. R. Soc. Open Sci. 4, 171153 (2017)CrossRefGoogle Scholar
  30. 30.
    Golaszewski, M., Grygoruk, R., Giorgio, I., Laudato, M., Di Cosmo, F.: Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions. Contin. Mech. Thermodyn. (to appear) (2018)Google Scholar
  31. 31.
    Turco, E., Barcz, K., Pawlikowski, M., Rizzi, N.L.: Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations. Zeitschrift für Angewandte Mathematik und Physik 67(122), 1–16 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Turco, E., Barcz, K., Rizzi, N.L.: Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part II: comparison with experimental evidence. Zeitschrift für Angewandte Mathematik und Physik 67(123), 1–16 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Turco, E., Golaszewski, M., Giorgio, I., D’Annibale, F.: Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Compos. Part B Eng. 118, 1–14 (2017)CrossRefGoogle Scholar
  34. 34.
    Giorgio, I.: Numerical identification procedure between a micro Cauchy model and a macro second gradient model for planar pantographic structures. Zeitschrift für Angewandte Mathematik und Mechanik 67(95), 1–17 (2016)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Placidi, L., Andreaus, U., Della Corte, A., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 66(6), 3699–3725 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Complex Syst. 3(3), 285–308 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets. Asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Misra, A., Poorsolhjouy, P.: Elastic behavior of 2D grain packing modeled as micromorphic media based on granular micromechanics. J. Eng. Mech. 143(1), C4016005 (2017)CrossRefGoogle Scholar
  39. 39.
    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)CrossRefGoogle Scholar
  40. 40.
    Riks, E.: The application of Newton’s method to the problem of elastic stability. J. Appl. Mech. Trans. ASME 39 Ser E(4), 1060–1065 (1972)ADSCrossRefzbMATHGoogle Scholar
  41. 41.
    Clarke, M.J., Hancock, G.J.: A study of incremental-iterative strategies for non-linear analyses. Int. J. Numer. Methods Eng. 29, 1365–1391 (1990)CrossRefGoogle Scholar
  42. 42.
    Turco, E., Caracciolo, P.: Elasto-plastic analysis of Kirchhoff plates by high simplicity finite elements. Comput. Methods Appl. Mech. Eng. 190, 691–706 (2000)ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Turco, E.: Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives. Eur. J. Environ. Civ. Eng. 21(5), 509–554 (2017)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Turco, E., dell’Isola, F., Rizzi, N.L., Grygoruk, R., Müller, W.H., Liebold, C.: Fiber rupture in sheared planar pantographic sheets: numerical and experimental evidence. Mech. Res. Commun. 76, 86–90 (2016)CrossRefGoogle Scholar
  45. 45.
    Turco, E., Rizzi, N.L.: Pantographic structures presenting statistically distributed defects: numerical investigations of the effects on deformation fields. Mech. Res. Commun. 77, 65–69 (2016)CrossRefGoogle Scholar
  46. 46.
    Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)CrossRefGoogle Scholar
  47. 47.
    Engelbrecht, J., Berezovski, A.: Reflections on mathematical models of deformation waves in elastic microstructured solids. Math. Mech. Complex Syst. 3(1), 43–82 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    dell’Isola, F., Forest, S.: Second gradient and generalized continua. A workshop held on 12-16 March 2012 in Cisterna di Latina. ZAMM J. Appl. Math. Mech. /Zeitschrift für Angewandte Mathematik und Mechanik 94(5), 367–372 (2014)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Placidi, L., Giorgio, I., Della Corte, A., Scerrato, D.: Euromech 563 Cisterna di Latina 17-21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: a review of presentations and discussions. Math. Mech. Solids 22(2), 144–157 (2017)CrossRefzbMATHGoogle Scholar
  50. 50.
    Aristodemo, M.: A high-continuity finite element model for two-dimensional elastic problems. Comput. Struct. 21(5), 987–993 (1985)CrossRefzbMATHGoogle Scholar
  51. 51.
    Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  52. 52.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Hoboken (2009)CrossRefzbMATHGoogle Scholar
  53. 53.
    Greco, L., Cuomo, M., Contraffatto, L., Gazzo, S.: An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Comput. Methods Appl. Mech. Eng. 324, 476–511 (2017)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Greco, L., Cuomo, M.: B-Spline interpolation of Kirchhoff-Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    dell’Isola, F., Maier, G., Perego, U., Andreaus, U., Esposito, R., Forest, S.: The complete works of Gabrio Piola: Volume I - Commented English Translation. Springer, Berlin (2014)zbMATHGoogle Scholar
  56. 56.
    De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Potts models in the continuum uniqueness and exponential decay in the restricted ensembles. J. Stat. Phys. 133(2), 281–345 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Coexistence of ordered and disordered phases in Potts models in the continuum. J. Stat. Phys. 134(2), 243–306 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Grimmett, G.R.: Correlation inequalities for the Potts model. Math. Mech. Complex Syst. 4(3–4), 327–334 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Diyaroglu, C., Oterkus, E., Oterkus, S., Madenci, E.: Peridynamics for bending of beams and plates with transverse shear deformation. Int. J. Solids Struct. 69, 152–168 (2015)CrossRefGoogle Scholar
  60. 60.
    De Meo, D., Diyaroglu, C., Zhu, N., Oterkus, E., Siddiq, M.A.: Modelling of stress-corrosion cracking by using peridynamics. Int. J. Hydrog. Energy 41(15), 6593–6609 (2016)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Architecture, Design and Urban planning (DADU)University of SassariSassariItaly
  2. 2.Civil, Environmental and Architectural Engineering (CEAE)The University of KansasLawrenceUSA
  3. 3.Mechanical Engineering (ME)The University of KansasLawrenceUSA
  4. 4.Warsaw University of TechnologyWarsawPoland
  5. 5.Department of Experimental Physiology and PathophysiologyMedical University of WarsawWarsawPoland

Personalised recommendations