Abstract
We describe a novel mechanical model of planar Timoshenko beam for large displacements analysis in elastic regime following Hencky beam model guidelines. More precisely, we model the strain energy of the beam in a discrete form by considering, besides the bending contribution, both the stretching and the sliding contributions. In this way a discrete model of Timoshenko beam is generated. This model, besides to be interesting di per sé has strong applications in the study of metamaterials based on beam lattices where, sometimes, the approximations introduced by the use of Euler–Bernoulli beam model are too rough for capturing some desired details. In addition, this is an intermediate step toward the construction of discrete three-dimensional Timoshenko beam models.
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Abali BE, Müller WH, Eremeyev VA (2015) Strain gradient elasticity with geometric nonlinearities and its computational evaluation. Mechanics of Advanced Materials and Modern Processes 1(1):1–11
Abali BE, Wu CC, Müller WH (2016) An energy-based method to determine material constants in nonlinear rheology with applications. Continuum Mechanics and Thermodynamics 28(5):1221–1246
Abdoul-Anziz H, Seppecher P (2018) Strain gradient and generalized continua obtained by homogenizing frame lattices. Mathematics and mechanics of complex systems 6(3):213–250
Abdoul-Anziz H, Seppecher P, Bellis C (2019) Homogenization of frame lattices leading to second gradient models coupling classical strain and strain-gradient terms. Mathematics and Mechanics of Solids 24(12):3976–3999
Alibert JJ, Seppecher P, dell’Isola F (2003) Truss modular beams with deformation energy depending on higher displacement gradients. Mathematics and Mechanics of Solids 8(1):51–73
Andreaus U, Spagnuolo M, Lekszycki T, Eugster SR (2018) A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams. Continuum Mechanics and Thermodynamics 30(5):1103–1123
Balobanov V, Niiranen J (2018) Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity. Computer Methods in Applied Mechanics and Engineering 339:137–159
Barchiesi E, Eugster SR, Placidi L, dell’Isola F (2019a) Pantographic beam: a complete second gradient 1d-continuum in plane. Zeitschrift für angewandte Mathematik und Physik 70(5):135
Barchiesi E, Ganzosch G, Liebold C, Placidi L, Grygoruk R, Müller WH (2019b) Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Continuum Mechanics and Thermodynamics 31(1):33–45
Boutin C, dell’Isola F, Giorgio I, Placidi L (2017) Linear pantographic sheets: asymptotic micromacro models identification. Mathematics and Mechanics of Complex Systems 5(2):127–162
Cazzani A, Stochino F, Turco E (2016a) An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik 96(10):1220–1244
Cazzani A, Stochino F, Turco E (2016b) On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation. Zeitschrift für angewandte Mathematik und Physik 67(2):24
Cazzani A, Stochino F, Turco E (2016c) On the whole spectrum of Timoshenko beams. Part II: further applications. Zeitschrift für angewandte Mathematik und Physik 67(2):25
Chróscielewski J, Schmidt R, Eremeyev VA (2019) Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches. Continuum Mechanics and Thermodynamics 31(1):147–188
Clarke MJ, Hancock GJ (1990) A study of incremental-iterative strategies for non-linear analyses. International Journal for Numerical Methods in Engineering 29(7):1365–1391
De Angelo M, Barchiesi E, Giorgio I, Abali BE (2019) Numerical identification of constitutive parameters in reduced-order bi-dimensional models for pantographic structures: application to out-of-plane buckling. Archive of Applied Mechanics 89(7):1333–1358
Della Corte A, Battista A, dell’Isola F, Seppecher P (2019) Large deformations of Timoshenko and Euler beams under distributed load. Zeitschrift für angewandte Mathematik und Physik 70(2):52
dell’Isola F, Andreaus U, Placidi L (2015) At the origins and in the vanguard of peridynamics, nonlocal and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Mathematics and Mechanics of Solids 20(8):887–928
dell’Isola F, Seppecher P, Alibert JJ, et al (2019a) Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mechanics and Thermodynamics 31(4):851–884
dell’Isola F, Seppecher P, Spagnuolo M, et al (2019b) Advances in pantographic structures: design, manufacturing, models, experiments and image analyses. Continuum Mechanics and Thermodynamics 31(4):1231–1282
Elishakoff I (2020) Who developed the so-called Timoshenko beam theory? Mathematics and Mechanics of Solids 25(1):97–116
Eremeev VA, Zubov LM (1994) On the stability of elastic of elastic bodies with couple stresses. Mechanics of Solids 29(3):172–181
EremeyevV, AltenbachH(2017) Basics of mechanics of micropolar shells. In: Shell-like Structures, vol 572, Springer, pp 63–111
Eremeyev VA (2019) Two-and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids. Acta Mechanica 230(11):3875–3887
Eugster SR, dell’Isola F (2017) Exegesis of the introduction and Sect. I from “Fundamentals of the mechanics of continua”** by E. Hellinger. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik 97(4):477–506
Eugster SR, dell’Isola F (2018a) Exegesis of Sect. II and III.Afrom “Fundamentals of the mechanics of continua” by E. Hellinger. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik 98(1):31–68
Eugster SR, dell’Isola F (2018b) Exegesis of Sect. III. B from “Fundamentals of the mechanics of continua” by E. Hellinger. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik 98(1):69–105
Fu YB, Ogden RW (1999) Nonlinear stability analysis of pre-stressed elastic bodies. Continuum Mechanics and Thermodynamics 11(3):141–172
Giorgio I (2020) A discrete formulation of Kirchhoff rods in large-motion dynamics. Mathematics and Mechanics of Solids 25(5):1081–1100
Giorgio I, Del Vescovo D (2018) Non-linear lumped-parameter modeling of planar multi-link manipulators with highly flexible arms. Robotics 7(4):60
Giorgio I, Del Vescovo D (2019) Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators. Mathematics and Mechanics of Complex Systems 7(2):159–174
Giorgio I, Della Corte A, dell’Isola F (2017) Dynamics of 1D nonlinear pantographic continua. Nonlinear Dynamics 88(1):21–31
Giorgio I, dell’Isola F, Steigmann DJ (2018) Axisymmetric deformations of a 2nd grade elastic cylinder. Mechanics Research Communications 94:45–48
Giorgio I, dell’Isola F, Steigmann DJ (2019) Edge effects in hypar nets. Comptes Rendus Mécanique 347(2):114–123
Greco L (2020) An iso-parametric G1-conforming finite element for the nonlinear analysis of Kirchhoff rod. Part I: the 2D case. Continuum Mechanics and Thermodynamics pp 1–24
Greco L, CuomoM(2015) Consistent tangent operator for an exact Kirchhoff rod model. Continuum Mechanics and Thermodynamics 27(4-5):861–877
Greco L, Cuomo M, Contrafatto L (2018) A reconstructed local B formulation for isogeometric Kirchhoff–Love shells. Computer method in applied mechanics and engineering 332:462–487
Greco L, Cuomo M, Contrafatto L (2019a) A quadrilateral G1-conforming finite element for the Kirchhoff plate model. Computer Methods in Applied Mechanics and Engineering 346:913–951
Greco L, Cuomo M, Contrafatto L (2019b) Two new triangular G1-conforming finite elements with cubic edge rotation for the analysis of Kirchhoff plates. Computer Methods in Applied Mechanics and Engineering 356:354–386
Gross A, Pantidis P, Bertoldi K, Gerasimidis S (2019) Correlation between topology and elastic properties of imperfect truss-lattice materials. Journal of the Mechanics and Physics of Solids 124:577–598
Hencky H (1921) Über die angenäherte lösung von stabilitätsproblemen im raum mittels der elastischen gelenkkette. PhD thesis, Engelmann
Kiendl J, Auricchio F, Hughes TJ, Reali A (2015) Single-variable formulations and isogeometric discretizations for shear deformable beams. Computer Methods in Applied Mechanics and Engineering 284:988–1004
Lakes RS (2018) Stability of Cosserat solids: size effects, ellipticity andwaves. Journal of Mechanics of Materials and Structures 13(1):83–91
Laudato M, Barchiesi E (2019) Non-linear dynamics of pantographic fabrics: modelling and numerical study. In: Wave Dynamics, Mechanics and Physics of Microstructured Metamaterials, Springer, pp 241–254
Luu AT, Kim NI, Lee J (2015) Isogeometric vibration analysis of free-form Timoshenko curved beams. Meccanica 50(1):169–187
Meza LR, Das S, Greer JR (2014) Strong, lightweight, and recoverable three-dimensional ceramic nanolattices. Science 345(6202):1322–1326
Meza LR, Phlipot GP, Portela CM, et al (2017) Reexamining the mechanical property space of three-dimensional lattice architectures. Acta Materialia 140:424–432
Misra A, Poorsolhjouy P (2015) Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Mathematics and Mechanics of Complex Systems 3(3):285–308
Misra A, Lekszycki T, Giorgio I, Ganzosch G, Müller WH, dell’Isola F (2018) Pantographic metamaterials show atypical poynting effect reversal. Mechanics Research Communications 89:6–10
Niiranen J, Balobanov V, Kiendl J, Hosseini SB (2019) Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro-and nano-beam models. Mathematics and Mechanics of Solids 24(1):312–335
Ogden RW (1997) Non-linear elastic deformations. Courier Corporation, Dover, Mineola
Pideri C, Seppecher P (1997) A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mechanics and Thermodynamics 9(5):241–257
Placidi L, dell’Isola F, Barchiesi E (2020) Heuristic homogenization of Euler and pantographic beams. In: Mechanics of Fibrous Materials and Applications, Springer, pp 123–155
Riks E (1972) The application of Newton’s method to the problem of elastic stability. Journal of Applied Mechanics, Transactions ASME E(4):1060–1065
Rodrigues O (1840) Des lois géométriques qui régissent les déplacements d’un système solide dans l’espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire. Journal de mathématiques pure at appliquées 1(5):380–440
Scerrato D, Giorgio I (2019) Equilibrium of two-dimensional cycloidal pantographic metamaterials in three-dimensional deformations. Symmetry 11(12):1523
Seppecher P, Alibert JJ, dell’Isola F (2011) Linear elastic trusses leading to continua with exotic mechanical interactions. In: Journal of Physics: Conference Series, IOP Publishing, vol 319, p 012018
Sheydakov DN, Altenbach H (2016) Stability of inhomogeneous micropolar cylindrical tube subject to combined loads. Mathematics and Mechanics of Solids 21(9):1082–1094
Solyaev Y, Lurie S, Barchiesi E, Placidi L (2020) On the dependence of standard and gradient elastic material constants on a field of defects. Mathematics and Mechanics of Solids 25(1):35–45
Spagnuolo M, Andreaus U (2019) A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling. Mathematics and Mechanics of Solids 24(1):258–280
Turco E (2018) Discrete is it enough? The revival of Piola–Hencky keynotes to analyze threedimensional Elastica. Continuum Mechanics and Thermodynamics 30(5):1039–1057
Turco E, dell’Isola F, Cazzani A, Rizzi NL (2016) Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4):85
Turco E, Misra A, Sarikaya R, Lekszycki T (2019) Quantitative analysis of deformation mechanisms in pantographic substructures: experiments and modeling. Continuum Mechanics and Thermodynamics 31(1):209–223
Turco E, Barchiesi E, Giorgio I, dell’Isola F (2020) A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory. International Journal of Non-Linear Mechanics 123:103481
Vangelatos Z, Komvopoulos K, Grigoropoulos C (2019a) Vacancies for controlling the behavior of microstructured three-dimensional mechanical metamaterials. Mathematics and Mechanics of Solids 24(2):511–524
Vangelatos Z, Melissinaki V, Farsari M, Komvopoulos K, Grigoropoulos C (2019b) Intertwined microlattices greatly enhance the performance of mechanical metamaterials. Mathematics and Mechanics of Solids 24(8):2636–2648
Wriggers P (2008) Nonlinear finite element methods. Springer Science & Business Media
Yang H, Ganzosch G, Giorgio I, Abali BE (2018) Material characterization and computations of a polymeric metamaterial with a pantographic substructure. Zeitschrift für angewandte Mathematik und Physik 69(4):105
Acknowledgements
For this work I am indebted to many people of the M&MOCS International Research Center. I wish to thank all of them for their invaluable suggestions and fruitful discussions.
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Turco, E. (2020). Modelling of Two-dimensional Timoshenko Beams in Hencky Fashion. In: Abali, B., Giorgio, I. (eds) Developments and Novel Approaches in Nonlinear Solid Body Mechanics. Advanced Structured Materials, vol 130. Springer, Cham. https://doi.org/10.1007/978-3-030-50460-1_11
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