In the present paper we consider a structure constituted by ‘long’ Euler beams forming two mutually intersecting arrays and interacting via internal pivots, which we call pantographic 2D lattices. For this structure, small deformations, but possibly large displacements, can be considered. We performed numerical simulations concerning 2D pantographic sheets of rectangular shape with two families of beam arrays cutting at 90 degrees. The set of theoretical tools needed to describe the continuous limit for such kind of structures goes beyond classical continuum mechanics. In particular, non-Cauchy contact actions can arise in the considered context. The results motivate further investigations, the first step reasonably being the determination of a generalized continuum model.
1. Altenbach, H. and Eremeyev, V. A. Analysis of the viscoelastic behavior of plates made of functionally graded materials. Z. angew. Math. Mech., 2008, 88(5), 332–341.
http://dx.doi.org/10.1002/zamm.200800001
2. Bîrsan, M., Altenbach, H., Sadowski, T., Eremeyev, V. A., and Pietras, D. Deformation analysis of functionally graded beams by the direct approach. Compos. Part B-Eng., 2012, 43(3), 1315–1328.
http://dx.doi.org/10.1016/j.compositesb.2011.09.003
3. Cao, J., Akkerman, R., Boisse, P., Chen, J., Cheng, H. S., de Graaf, E. F. et al. Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results. Compos. Part A-Appl.S., 2008, 39(6), 1037–1053.
4. Carcaterra, A. and Akay, A. Dissipation in a finite-size bath. Phys. Rev. E, 2011, 84, 011121.
http://dx.doi.org/10.1103/PhysRevE.84.011121
5. Carcaterra, A., Roveri, N., and Pepe, G. Fractional dissipation generated by hidden wave-fields. Math. Mech. Solids, 2014.
http://dx.doi.org/10.1177/1081286513518941
6. Cazzani, A. On the dynamics of a beam partially supported by an elastic foundation: an exact solution-set. Int. J. Str. Stab. Dyn., 2013, 13(08), 1350045.
http://dx.doi.org/10.1142/S0219455413500454
7. Cazzani, A. and Ruge, P. Numerical aspects of coupling strongly frequency-dependent soil-foundation models with structural finite elements in the time-domain. Soil Dyn. Earthq. Eng., 2012, 37, 56–72.
http://dx.doi.org/10.1016/j.soildyn.2012.01.011
8. Cazzani, A., Malagù, M., and Turco, E. Isogeometric analysis of plane-curved beams. Math. Mech. Solids, 2014.
http://dx.doi.org/10.1177/1081286514531265
9. Cuomo, M., Contrafatto, L., and Greco, L. A variational model based on isogeometric interpolation for the analysis of cracked bodies. Int. J. Eng. Sci., 2014, 80, 173–188.
http://dx.doi.org/10.1016/j.ijengsci.2014.02.017
10. Dos Reis, F. and Ganghoffer, J. F. Equivalent mechanical properties of auxetic lattices from discrete homogenization. Comput. Mater. Sci., 2012, 51, 314–321.
http://dx.doi.org/10.1016/j.commatsci.2011.07.014
11. Eremeyev, V. A. and Pietraszkiewicz, W. Phase transitions in thermoelastic and thermoviscoelastic shells. Arch. Mech., 2009, 61(1), 41–67.
12. Eremeyev, V. A. and Zubov, L. M. On the stability of equilibrium of nonlinear elastic bodies with phase transformations. Proc. USSR Acad. Sci. Mech. Solids, 1991, 2, 56–65.
13. Fornberg, B. A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge, 1998.
14. Garusi, E., Tralli, A., and Cazzani, A. An unsymmetric stress formulation for Reissner–Mindlin plates: a simple and locking-free rectangular element. Int. J. Comp. Eng. Sci., 2004, 5(3), 589–618.
http://dx.doi.org/10.1142/S1465876304002587
15. Ghiba, I. D., Neff, P., Madeo, A., Placidi, L., and Rosi, G. The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics. Math. Mech. Solids, 2014. doi: 1081286513516972.
16. Goda, I., Assidi, M., Belouettar, S., and Ganghoffer, J. F. A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. J. Mech. Behav. Biomed. Mater., 2012, 16, 87–108.
http://dx.doi.org/10.1016/j.jmbbm.2012.07.012
17. Greco, L. and Cuomo, M. B-Spline interpolation of Kirchhoff–Love space rods. Comput. Methods Appl. Mech. Eng., 2013, 256, 251–269.
http://dx.doi.org/10.1016/j.cma.2012.11.017
18. Greco, L. and Cuomo, M. An implicit G1 multi patch B-spline interpolation for Kirchhoff–Love space rod. Comput. Methods Appl. Mech. Eng., 2014, 269, 173–197.
http://dx.doi.org/10.1016/j.cma.2013.09.018
19. Greco, L., Impollonia, N., and Cuomo, M. A procedure for the static analysis of cable structures following elastic catenary theory. Int. J. Solids Struct., 2014, 51(7), 1521–1533.
http://dx.doi.org/10.1016/j.ijsolstr.2014.01.001
20. Luongo, A. and Piccardo, G. Linear instability mechanisms for coupled translational galloping. J. Sound Vib., 2005, 288(4), 1027–1047.
http://dx.doi.org/10.1016/j.jsv.2005.01.056
21. Luongo, A. and Zulli, D. Dynamic instability of inclined cables under combined wind flow and support motion. Nonlinear Dynam., 2012, 67(1), 71–87.
http://dx.doi.org/10.1007/s11071-011-9958-9
22. Luongo, A., Paolone, A., and Piccardo, G. Postcritical behavior of cables undergoing two simultaneous galloping modes. Meccanica, 1998, 33(3), 229–242.
http://dx.doi.org/10.1023/A:1004343029604
23. Luongo, A., Zulli, D., and Piccardo, G. A linear curved-beam model for the analysis of galloping in suspended cables. J. Mech. Mater. Struct., 2007, 2(4), 675–694.
http://dx.doi.org/10.2140/jomms.2007.2.675
24. Madeo, A., Neff, P., Ghiba, I. D., Placidi, L., and Rosi, G. Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Continuum Mech. Therm., 2013.
http://dx.doi.org/10.1007/s00161-013-0329-2
25. Madeo, A., Neff, P., Ghiba, I. D., Placidi, L., and Rosi, G. Band gaps in the relaxed linear micromorphic continuum. Z. angew. Math. Mech., 2014.
http://dx.doi.org/10.1002/zamm.201400036
26. Maugin, G. A. Solitons in elastic solids (1938–2010). Mech. Res. Commun., 2011, 38(5), 341–349.
http://dx.doi.org/10.1016/j.mechrescom.2011.04.009
27. Nahine, H. and Boisse, P. Tension locking in finite-element analyses of textile composite reinforcement deformation. C.R. Acad. Sci., Ser. IIb: Mec., Phys., 2013, 341(6), 508–519.
28. Nahine, H. and Boisse, P. Locking in simulation of composite reinforcement deformations. Analysis and treatment. Compos. Part A-Appl.S., 2013, 53, 109–117.
29. Neff, P., Ghiba, I.-D., Madeo, A., Placidi, L., and Rosi, G. A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech. Therm., 2013, 26(5), 639–681.
http://dx.doi.org/10.1007/s00161-013-0322-9
30. Piccardo, G., Pagnini, L. C., and Tubino, F. Some research perspectives in galloping phenomena: critical conditions and post-critical behavior. Continuum Mech. Therm., 2014, 27(5), 261–285.
31. Pideri, C. and Seppecher, P. A second gradient material resulting from the homogenization of an hetero\-geneous linear elastic medium. Continuum Mech. Therm., 1997, 9(5), 241–257.
http://dx.doi.org/10.1007/s001610050069
32. Pideri, C. and Seppecher, P. Asymptotics of a non-planar rod in non-linear elasticity. Asymptotic Anal., 2006, 48(1–2), 33–54.
33. Pietraszkiewicz, W., Eremeyev, V., and Konopinska, V. Extended non-linear relations of elastic shells undergoing phase transitions. Z. angew. Math. Mech., 2007, 87(2), 150–159.
http://dx.doi.org/10.1002/zamm.200610309
34. Salupere, A. The pseudospectral method and discrete spectral analysis. In Applied Wave Mathematics: Selected Topics in Solids, Fluids, and Mathematical Methods (Quak, E. and Soomere, T., eds). Springer, Berlin, 2009, 301–333.
http://dx.doi.org/10.1007/978-3-642-00585-5_16
35. Sestieri, A. and Carcaterra, A. Space average and wave interference in vibration conductivity. J. Sound Vib., 2003, 263(3), 475–491.
http://dx.doi.org/10.1016/S0022-460X(02)01060-X
36. Tarantino, G. Esempi di utilizzo di differenti ambienti di modellizzazione per lo studio delle onde meccaniche. Quaderni di Ricerca in Didattica (Science), 2010, No. 1, G.R.I.M. (Department of Mathematics, University of Palermo, Italy).
37. Turco, E. and Caracciolo, P. Elasto-plastic analysis of Kirchhoff plates by high simplicity finite elements. Comput. Methods Appl. Mech. Eng., 2000, 190(5–7), 691–706.
http://dx.doi.org/10.1016/S0045-7825(99)00438-7
38. Yang, Y. and Misra, A. Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation. Comput. Model. Eng. Sci., 2010, 64, 1–36.
39. Yang, Y., Ching, W. Y., and Misra, A. Higher-order continuum theory applied to fracture simulation of nano-scale intergranular glassy film. J. Nanomech. Micromech., 2011, 1, 60–71.
http://dx.doi.org/10.1061/(ASCE)NM.2153-5477.0000030
40. Yeremeyev, V. A., Freidin, A. B., and Sharipova, L. L. Non-uniqueness and stability in problems of the equilibrium of elastic two-phase solids. Dokl. Ross. Akad. Nauk, 2003, 391(2), 189–193.
41. Zabusky, N. J. and Kruskal, M. D. Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 1965, 15, 240–243.http://dx.doi.org/10.1103/PhysRevLett.15.240
