×

Modeling of three-dimensional beam nonlinear vibrations generalizing Hencky’s ideas. (English) Zbl 07619117

Summary: In this contribution, a novel nonlinear micropolar beam model suitable for metamaterials design in a dynamics framework is presented and discussed. The beam model is formulated following a completely discrete approach and it is fully defined by its Lagrangian, i.e., by the kinetic energy and by the potential of conservative forces. Differently from Hencky’s seminal work, which considers only flexibility to compute the buckling load for rectilinear and planar Euler-Bernoulli beams, the proposed model is fully three-dimensional and considers both the extensional and shear deformability contributions to the strain energy and translational and rotational kinetic energy terms. After having introduced the model formulation, some simulations obtained with a numerical integration scheme are presented to show the capabilities of the proposed beam model.

MSC:

74-XX Mechanics of deformable solids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dell’Isola F, Seppecher P, Spagnuolo M, et al. Advances in pantographic structures: design, manufacturing, models, experiments and image analyses. Continuum Mech Thermodyn2019; 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 Thermodyn2019; 31(4): 851-884.
[3] Barchiesi E, dell’Isola F, Bersani AM, et al. Equilibria determination of elastic articulated duoskelion beams in 2D via a Riks-type algorithm. Int J Non-Linear Mech2021; 128: 103628.
[4] Turco E, dell’Isola F, Cazzani A, et al. Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift Angewandte Math Physik2016; 67(85): 1-28. · Zbl 1432.74158
[5] Giorgio I, Rizzi NL, Turco E. Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc Royal Soc A: Math Phys Eng Sci2017; 473(2207): 1-21. · Zbl 1404.74064
[6] Turco E. Discrete is it enough? The revival of Piola-Hencky keynotes to analyze three-dimensional. Elastica Continuum Ech Thermodyn2018; 30(5): 1039-1057. · Zbl 1396.74008
[7] 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 Mech2020; 123: 103481.
[8] Andreaus U, Spagnuolo M, Lekszycki T, et al. A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams. Continuum Mech Thermodyn2018; 30: 1103-1123. · Zbl 1396.74070
[9] Elishakoff I. Who developed the so-called Timoshenko beam theory? Math Mech Solids2019; 25: 97-116. · Zbl 1446.74003
[10] 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 Royal Soc Lond A: Math Hys Eng Sci2016; 472(2185): 1-23.
[11] Barchiesi E, Laudato M, Di Cosmo F. Wave dispersion in non-linear pantographic beams. Mech Res Commun2018; 94: 128-132.
[12] 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. Zamm-Zeitschrift Ngewandte Math Mech2019; 99(7): 1-20. · Zbl 07785932
[13] Giorgio I. A discrete formulation of Kirchhoff rods in large-motion dynamics. Math Mech Solids2020; 25: 1081-1100. · Zbl 1482.74105
[14] 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 Struct2018; 147: 94-109.
[15] Hencky H. Über die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. PhD Thesis, Engelmann, 1921.
[16] Barchiesi E, Ganzosch G, Liebold C, et al. Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Continuum Mech Thermodyn2019; 31: 33-45.
[17] Desmorat B, Spagnuolo M, Turco E. Stiffness optimization in nonlinear pantographic structures. Math Mech Solids2020; 25(11): 2252-2262. · Zbl 1482.74143
[18] Dell’Isola F, Steigmann D, Della Corte A, et al. Discrete and continuum models for complex metamaterials. Cambridge: Cambridge University Press, 2020.
[19] Bersani AM, Caressa P. Lagrangian description of dissipative systems: a review. Math Mech Solids2020; 26: 785-803. · Zbl 07357428
[20] Luongo A, Zulli D. Mathematical models of beams and cables. New York: John Wiley & Sons, 2013.
[21] Eremeyev VA, Altenbach H. Basics of mechanics of micropolar shells. In: Altenbach H, Eremeyev VA (eds) Shell-like structures, vol. 572. Cham: Springer, 2017, pp. 63-111.
[22] Wriggers P. Nonlinear finite element methods. Cham: Springer, 2008. · Zbl 1153.74001
[23] Casciaro R. Time evolutional analysis of nonlinear structures. Meccanica1975; 3(X): 156-167. · Zbl 0374.73069
[24] Turco E. Stepwise analysis of pantographic beams subjected to impulsive loads. Math Mech Solids2020; 26: 62-79. · Zbl 1486.74088
[25] Greco L, Cuomo M, Contrafatto L. A reconstructed local \(B\) formulation for isogeometric Kirchhoff-Love shells. Comput Method Appl Mech Eng2018; 332: 462-487. · Zbl 1440.74395
[26] Cazzani A, Malagù M, Turco E. Isogeometric analysis of plane curved beams. Math Mech Solids2016; 21(5): 562-577. · Zbl 1370.74084
[27] Greco L. An iso-parametric G1—conforming finite element for the nonlinear analysis of Kirchhoff rod—part I: the 2D case. Continuum Mech Thermodyn2020; 32: 1473-1496.
[28] Riks E. The application of Newton’s method to the problem of elastic stability. J Appl Mech Transactions ASME1972; 39(4): 1060-1065. · Zbl 0254.73047
[29] Hutchinson JW, Budiansky B. Dynamic buckling estimates. AIAA J1966; 4(3): 525-530.
[30] Cazzani A, Stochino F, Turco E. On the whole spectrum of Timoshenko beams—part I: a theoretical revisitation. Zeitschrift Angewandte Math Physik2016; 67(24): 1-30. · Zbl 1338.74069
[31] Cazzani A, Stochino F, Turco E. On the whole spectrum of Timoshenko beams—part II: further applications. Zeitschrift Angewandte Math Physik2016; 67(25): 1-21. · Zbl 1338.74068
[32] Bauer AM, Breitenberger M, Philipp B, et al. Nonlinear isogeometric spatial Bernoulli beam. Comput Methods Appl Mech Eng2016; 303: 101-127. · Zbl 1425.74450
[33] Giorgio I, Scerrato D. Multi-scale concrete model with rate-dependent internal friction. J Environ Civ Eng2017; 21(7-8): 821-839.
[34] Ciallella A, Pasquali D, Golaszewski M, et al. A rate-independent internal friction to describe the hysteretic behavior of pantographic structures under cyclic loads. Mech Res Commun2021; 2021: 103761.
[35] Dell’Isola F, Lekszycki T, Pawlikowski M, et al. Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Zeitschrift Angewandte Math Physik2015; 66(6): 3473-3498. · Zbl 1395.74002
[36] Giorgio I, dell’Isola F, Misra A. Chirality in 2D Cosserat media related to stretch-micro-rotation coupling with links to granular micromechanics. Int J Solids Struct2020; 202: 28-38.
[37] Ganzosch G, Hoschke K, Lekszycki T, et al. 3D-measurements of 3D-deformations of pantographic structures. Technische Mechanik2018; 38(3): 233-245.
[38] Dell’Isola F, Turco E, Misra A, et al. Force-displacement relationship in micro-metric pantographs: experiments and numerical simulations. Compt Rend: Mécan2019; 347(5): 397-405.
[39] Vangelatos Z, Yildzdag ME, Giorgio I, et al. Investigating the mechanical response of microscale pantographic structures fabricated by multiphoton lithography. Extrem Mech Lett2021; 43: 101202.
[40] Yang H, Ganzosch G, Giorgio I, et al. Material characterization and computations of a polymeric metamaterial with a pantographic substructure. Zeitschrift Für Angewandte Math Und Physik2018; 69(4): 105. · Zbl 1401.74016
[41] Placidi L. A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model. Continuum Mech Thermodyn2016; 28(1-2): 119-137. · Zbl 1348.74062
[42] Placidi L, Misra A, Barchiesi E. Two-dimensional strain gradient damage modeling: a variational approach. Zeitschrift Angewandte Math Physik2018; 69(3): 56. · Zbl 1393.74168
[43] Barchiesi E, Eugster SR, dell’Isola F, et al. Large in-plane elastic deformations of bi-pantographic fabrics: asymptotic homogenization and experimental validation. Math Mech Solids2020; 25(3): 739-767. · Zbl 1446.74171
[44] Turco E. In-plane shear loading of granular membranes modeled as a Lagrangian assembly of rotating elastic particles. Mech Res Commun2018; 92: 61-66.
[45] Turco E, dell’Isola F, Misra A. A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations. Int J Numer Anal Methods Geomech2019; 43(5): 1051-1079.
[46] Falsone G, La Valle G. A homogenized theory for functionally graded Euler-Bernoulli and Timoshenko beams. Acta Mech2019; 230(10): 3511-3523. · Zbl 1428.74184
[47] Moler CB, Stewart GW. An algorithm for generalized matrix eigenvalue problems. SIAM J Numer Analysis1973; 10(2): 241-256. · Zbl 0253.65019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.