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Stepwise analysis of pantographic beams subjected to impulsive loads. (English) Zbl 1486.74088

Summary: Materials based on pantographic unit cells have very interesting mechanical peculiarities. For these reasons they are largely studied from a theoretical, experimental, and numerical point of view. Numerical simulations furnish an important contribution for the the design and optimization of such materials and, more generally, for metamaterials. Here, we consider the influence of inertial forces, removing the hypothesis of quasistatic loading. By using an intrinsically discrete model, inspired by Hencky’s ideas, already tested in a series of published works, here we add the contribution of inertial forces and, in the framework of stepwise schemes, we re-experience an adaptive integration scheme capable of reconstructing the best structural response corresponding to a prefixed time step. Several numerical simulations, although preparatory, inspire some remarks on materials based on pantographic cells and outline the way for future challenges.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] dell’Isola, F, Seppecher, P, Spagnuolo, M, et al. Advances in pantographic structures: Design, manufacturing, models, experiments and image analyses. Continuum Mech Thermodyn 2019; 31(4): 1231-1282.
[2] dell’Isola, F, Seppecher, P, Alibert, JJ, et al. Pantographic metamaterials: An example of mathematically driven design and of its technological challenges. Continuum Mech Thermodyn 2019; 31(4): 851-884.
[3] dell’Isola, F, Giorgio, I, Pawlikowski, M, et al. Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium. Proc R Soc London, Ser A 2016; 472(2185): 20150790.
[4] Giorgio, I, Rizzi, NL, Turco, E. Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc R Soc London, Ser A 2017; 473(2207): 20170636. · Zbl 1404.74064
[5] Giorgio, I, Della Corte, A, dell’Isola, F. Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn 2017; 88(1): 21-31.
[6] dell’Isola, F, Della Corte, A, Giorgio, I, et al. Pantographic 2D sheets: Discussion of some numerical investigations and potential applications. Int J Non Linear Mech 2016; 80: 200-208.
[7] dell’Isola, F, Giorgio, I, Andreaus, U. Elastic pantographic 2D lattices: A numerical analysis on static response and wave propagation. Proc Est Acad Sci 2015; 64(3): 219-225.
[8] Laudato, M, Barchiesi, E. Non-linear dynamics of pantographic fabrics: Modelling and numerical study. In: Wave dynamics, mechanics and physics of microstructured metamaterials. Cham: Springer, 2019, 241-254.
[9] Placidi, L, Barchiesi, E, Turco, E, et al. A review on 2D models for the description of pantographic fabrics. Z Angew Math Phys 2016; 67: 121. · Zbl 1359.74019
[10] Barchiesi, E, Placidi, L. A review on models for the 3D statics and 2D dynamics of pantographic fabrics. In: Sumbatyan, M (ed.) Wave dynamics and composite mechanics for microstructured materials and metamaterials (Advanced Structured Materials, vol. 59). Singapore: Springer, 2017, 239-258.
[11] Barchiesi, E, Spagnuolo, M, Placidi, L. Mechanical metamaterials: A state of the art. Math Mech Solids 2018; 24(1): 212-234. · Zbl 1425.74036
[12] Greco, L. An iso-parametric \(G^1\)-conforming finite element for the nonlinear analysis of Kirchhoff rod. Part I: The 2D case. Continuum Mech Thermodyn. Epub ahead of print 14 January 2020. DOI: 10.1007/s00161-020-00861-9.
[13] Laudato, M, Manzari, L, Barchiesi, E, et al. First experimental observation of the dynamical behavior of a pantographic metamaterial. Mech Res Commun 2018; 94: 125-127.
[14] Hencky, H. Über die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. Leipzig: W Engelmann, 1921.
[15] Turco, E, dell’Isola, F, Cazzani, A, et al. Hencky-type discrete model for pantographic structures: Numerical comparison with second gradient continuum models. Z Angew Math Phys 2016; 67: 85. · Zbl 1432.74158
[16] Turco, E, Golaszewski, M, Cazzani, A, et al. Large deformations induced in planar pantographic sheets by loads applied on fibers: Experimental validation of a discrete Lagrangian model. Mech Res Commun 2016; 76: 51-56.
[17] Baroudi, D, Giorgio, I, Battista, A, et al. Nonlinear dynamics of uniformly loaded elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations. Z Angew Math Mech 2019; 99(7): e201800121.
[18] Giorgio, I. A discrete formulation of Kirchhoff rods in large-motion dynamics. Math Mech Solids 2020; 25(5): 1081-1100. · Zbl 1482.74105
[19] Giorgio, I, Del Vescovo, D. Energy-based trajectory tracking and vibration control for multi-link highly flexible manipulators. Math Mech Complex Syst 2019; 7(2): 159-174. · Zbl 1458.70004
[20] Giorgio, I, Del Vescovo, D. Non-linear lumped-parameter modeling of planar multi-link manipulators with highly flexible arms. Robotics 2018; 7(4): 60.
[21] Wriggers, P . Solution methods for time dependent problems. In: Nonlinear finite element methods. Berlin: Springer, 2008, 205-254.
[22] Casciaro, R. Time evolutional analysis of nonlinear structures. Meccanica 1975; 3(X): 156-167. · Zbl 0374.73069
[23] Aristodemo, M. A high-continuity finite element model for two-dimensional elastic problems. Comput Struct 1985; 21(5): 987-993. · Zbl 0587.73109
[24] Bilotta, A, Formica, G, Turco, E. Performance of a high-continuity finite element in three-dimensional elasticity. Int J Numer Methods Biomed Eng 2010; 26: 1155-1175. · Zbl 1426.74282
[25] Turco, E, Barcz, K, Pawlikowski, M, et al. Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: Numerical simulations. Z Angew Math Phys 2016; 67(5): 122. · Zbl 1432.74156
[26] Turco, E, Barcz, K, Rizzi, NL. Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part II: Comparison with experimental evidence. Z Angew Math Phys 2016; 67(5): 123. · Zbl 1432.74157
[27] Turco, E, Rizzi, NL. Pantographic structures presenting statistically distributed defects: Numerical investigations of the effects on deformation fields. Mech Res Commun 2016; 77: 65-69.
[28] Turco, E, Barchiesi, E. Equilibrium paths of Hencky pantographic beams in a three-point bending problem. Math Mech Complex Syst 2019; 7(4): 287-310. · Zbl 1434.74076
[29] Turco, E, Misra, A, Pawlikowski, M, et al. Enhanced Piola-Hencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments. Int J Solids Struct 2018; 147: 94-109.
[30] Barchiesi, E, dell’Isola, F, Laudato, M, et al. A 1D continuum model for beams with pantographic microstructure: Asymptotic micro-macro identification and numerical results. Advances in mechanics of microstructured media and structures (Advanced Structured Materials vol. 87). Cham: Springer, 2018, 43-74.
[31] Barchiesi, E, Eugster, SR, Placidi, L, et al. Pantographic beam: A complete second gradient 1D-continuum in plane. Z Angew Math Phys 2019; 70(5): 135. · Zbl 1425.74387
[32] dell’Isola, F, Steigmann, D, Della Corte, A, et al. Discrete and continuum models for complex metamaterials. Cambridge: Cambridge University Press, 2020.
[33] Katsikadelis, JT. A new direct time integration method for the equations of motion in structural dynamics. Z Angew Math Mech 2013; 94(9): 757-774. · Zbl 1298.74251
[34] de Miranda, S, Mancuso, M, Ubertini, F. Time discontinuous Galerkin methods with energy decaying correction for non-linear elastodynamics. Int J Numer Methods Eng 2010; 83: 323-347. · Zbl 1193.74050
[35] Bathe, KJ, Noh, G. Insight into an implicit time integration scheme for structural dynamics. Comput Struct 2012; 98-99: 1-6.
[36] Wriggers, P. Nonlinear finite element methods. Berlin: Springer, 2008. · Zbl 1153.74001
[37] Casciaro, R. An optimal time discretization method in structural analysis. Technical Report 161, Istituto di Scienza delle costruzioni, Università degli Studi di Roma, 1974.
[38] Hughes, TJR . Analysis of transient algorithms with particular reference to stability behaviour. In: Belytschko, T, Hughes, TJR (eds.) Computational methods for transient analysis ( Computational Methods in Mechanics, vol. 1). Amsterdam: North-Holland, 1983, 67-155.
[39] Clough, RW, Penzien, J. Response to harmonic loading. In: Dynamics of structures. 3rd ed. Berkeley: Computers & Structures, Berkeley, 2003, 33-64.
[40] Riks, E. The application of Newton’s method to the problem of elastic stability. J Appl Mech 1972; 39(4): 1060-1065. · Zbl 0254.73047
[41] Nayfeh, AH, Mook, DT. Nonlinear oscillations. New York: Wiley, 1979. · Zbl 0418.70001
[42] Lacarbonara, W. Nonlinear structural mechanics: Theory, dynamical phenomena and modeling. New York: Springer, 2013. · Zbl 1263.74001
[43] Turco, E. Numerically driven tuning of equilibrium paths for pantographic beams. Continuum Mech Thermodyn 2019; 31(6): 1941-1960.
[44] Turco, E, Misra, A, Sarikaya, R, et al. Quantitative analysis of deformation mechanisms in pantographic substructures: Experiments and modeling. Continuum Mech Thermodyn 2019; 31(1): 209-223.
[45] dell’Isola, F, Turco, E, Misra, A, et al. Force-displacement relationship in micro-metric pantographs: Experiments and numerical simulations. C R Mec 2019; 347(5): 397-405.
[46] Turco, E, Barchiesi, E, Giorgio, I, et al. A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory. Int J Non Linear Mech 2020; 123: 103481.
[47] Turco, E. Discrete is it enough? The revival of Piola-Hencky keynotes to analyze three-dimensional Elastica. Continuum Mech Thermodyn 2018; 30(5): 1039-1057. · Zbl 1396.74008
[48] Altenbach, H, Eremeyev, VA, Lebedev, LP, et al. Acceleration waves and ellipticity in thermoelastic micropolar media. Arch Appl Mech 2010; 80(3): 217-227. · Zbl 1271.74251
[49] Altenbach, H, Eremeyev, VA. On nonlinear dynamic theory of thin plates with surface stresses. In: Contributions to advanced dynamics and continuum mechanics. Cham: Springer, 2019.
[50] Eremeyev, VA. Strongly anisotropic surface elasticity and antiplane surface waves. Philos Trans R Soc London, Ser A 2020; 378(2162): 20190100.
[51] Turco, E, Golaszewski, M, Giorgio, I, et al. Pantographic lattices with non-orthogonal fibres: Experiments and their numerical simulations. Composites, Part B 2017; 118: 1-14.
[52] Andreaus, U, Baragatti, P, Placidi, L. Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation. Int J Non Linear Mech 2016; 80: 96-106.
[53] Lekszycki, T, Olhoff, N, Pedersen, JJ. Modelling and identification of viscoelastic properties of vibrating sandwich beams. Compos Struct 1992; 22(1): 15-31.
[54] dell’Isola, F, Rosa, L, Wozniak, C. Dynamics of solids with microperiodic nonconnected fluid inclusions. Arch Appl Mech 1997; 67(4): 215-228. · Zbl 0888.73004
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