A nonlinear model for the dynamics of a Kirchhoff rod in the three-dimensional space is developed in the framework of a discrete elastic theory. The formulation avoids the use of Euler angles for the orientation of the rod cross-sections to provide a computationally singularity-free parameterization of rotations along the motion trajectories. The material directions related to the principal axes of the cross-sections are specified using auxiliary points that must satisfy constraints enforced by the Lagrange multipliers method. A generalization of this approach is presented to take into account Poisson’s effect in an orthotropic rod. Numerical simulations are performed to test the presented formulation.
EugsterSRHeschCBetschP, et al. Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. Int J Numer Meth Eng2014; 97(2): 111–129.
2.
GrecoLCuomoM.Consistent tangent operator for an exact Kirchhoff rod model. Continuum Mech Thermodyn2014; 27(4–5): 861–877.
3.
LuongoAZulliD.Mathematical models of beams and cables. New York: John Wiley & Sons, 2013.
4.
PideriCSeppecherP.Asymptotics of a non-planar rod in non-linear elasticity. Asymptotic Anal2006; 48(1, 2): 33–54.
5.
SteigmannDJFaulknerMG.Variational theory for spatial rods. J Elasticity1993; 33(1): 1–26.
6.
JawedMKNoveliaAO’ReillyOM.A primer on the kinematics of discrete elastic rods. New York: Springer, 2018.
7.
BergouMWardetzkyMRobinsonS, et al. Discrete elastic rods. ACM Trans Graphics27: 63.
8.
TurcoE.Discrete is it enough? The revival of Piola–Hencky keynotes to analyze three-dimensional Elastica. Continuum Mech Thermodyn2018; 30(5): 1039–1057.
9.
BrezovDSMladenovaCDMladenovIM. New perspective on the gimbal lock problem. In AIP Conference Proceedings, Vol. 1570. AIP, pp. 367–374.
10.
PaiPF.Geometrically exact beam theory without Euler angles. Int J Solids Struct2011; 48(21): 3075–3090.
11.
TurcoEdell’IsolaFCazzaniA, et al. Hencky-type discrete model for pantographic structures: Numerical comparison with second gradient continuum models. Z angew Math Phys2016; 67(4): 85.
12.
WangCMZhangHGaoRP, et al. Hencky bar-chain model for buckling and vibration of beams with elastic end restraints. Int J Struct Stability Dynam2015; 15(07): 1540007.
13.
ZhangHWangCMChallamelN.Buckling and vibration of Hencky bar-chain with internal elastic springs. Int J Mech Sci2016; 119: 383–395.
14.
ZhangHWangCMRuoccoE, et al. Hencky bar-chain model for buckling and vibration analyses of non-uniform beams on variable elastic foundation. Eng Struct2016; 126: 252–263.
15.
BaroudiDGiorgioIBattistaA, et al. Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations. Z angew Math Mech2019; 99: 7.
16.
ColemanBDDillEHLemboM, et al. On the dynamics of rods in the theory of Kirchhoff and Clebsch. Arch Rat Mech Anal1993; 121(4): 339–359.
17.
BersaniAdell’IsolaFSeppecherP.Lagrange multipliers in infinite dimensional spaces, examples of application. In: AltenbachHchsnerA (eds) Encyclopedia of Continuum Mechanics. Berlin: Springer, 2019.
18.
dell’IsolaFDi CosmoF.Lagrange multipliers in infinite-dimensional systems, methods of. In: AltenbachHchsnerA. (eds) Encyclopedia of Continuum Mechanics. Berlin: Springer, 2018.
19.
dell’IsolaFGiorgioIPawlikowskiM, et al. Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc A Math Phys Eng Sci2016; 472(2185): 20150790.
20.
dell’IsolaFSeppecherPAlibertJJ, et al. Pantographic metamaterials: An example of mathematically driven design and of its technological challenges. Continuum Mech Thermodyn2019; 31(4): 851–884.
21.
SimoJC. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput Meth Appl Mech Eng1985; 49(1): 55–70.
22.
SimoJCVu-QuocL.A three-dimensional finite-strain rod model. Part II: Computational aspects. Comput Meth Appl Mech Eng1986; 58(1): 79–116.
23.
SimoJCVu-QuocL.On the dynamics in space of rods undergoing large motions—a geometrically exact approach. Comput Meth Appl Mech Eng1988; 66(2): 125–161.
24.
AltenbachHEremeyevV.On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math Mech Complex Syst2015; 3(3): 273–283.
25.
LekszyckiTOlhoffNPedersenJJ.Modelling and identification of viscoelastic properties of vibrating sandwich beams. Composite Struct1992; 22(1): 15–31.
26.
AbaliBEWuCCMüllerWH.An energy-based method to determine material constants in nonlinear rheology with applications. Continuum Mech Thermodyn2016; 28(5): 1221–1246.
27.
FerrarottiAPiccardoGLuongoA.A novel straightforward dynamic approach for the evaluation of extensional modes within GBT ‘cross-section analysys’. Thin-Walled Struct2017; 114: 52–69.
PiccardoGRanziGLuongoA.A direct approach for the evaluation of the conventional modes within the GBT formulation. Thin-Walled Struct2014; 74: 133–145.
30.
Della CorteAdell’IsolaFEspositoR, et al. Equilibria of a clamped Euler beam (Elastica) with distributed load: Large deformations. Math Mod Meth Appl Sci2017; 27(08): 1391–1421.
31.
BarchiesiESpagnuoloMPlacidiL.Mechanical metamaterials: A state of the art. Math Mech Solids2019; 24(1): 212–234.
32.
dell’IsolaFAndreausUPlacidiL.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 Solids2015; 20(8): 887–928.
33.
AvellaMDell’ErbaRMartuscelliE.Fiber reinforced polypropylene: Influence of iPP molecular weight on morphology, crystallization, and thermal and mechanical properties. Polymer Composites1996; 17(2): 288–299.
34.
KarathanasopoulosNRedaHGanghofferJF.Designing two-dimensional metamaterials of controlled static and dynamic properties. Computat Mater Sci2017; 138: 323–332.
35.
NejadsadeghiNPlacidiLRomeoM, et al. Frequency band gaps in dielectric granular metamaterials modulated by electric field. Mech Res Commun2019; 95: 96–103.
36.
RedaHKarathanasopoulosNElnadyK, et al. Mechanics of metamaterials: An overview of recent developments. In Advances in Mechanics of Microstructured Media and Structures. Springer, 2018, pp. 273–296.
37.
VangelatosZKomvopoulosKGrigoropoulosCP.Vacancies for controlling the behavior of microstructured three-dimensional mechanical metamaterials. Math Mech Solids2019; 24(2): 511–524.
38.
SpagnuoloMFranciosiPdell’IsolaF.A Green operator-based elastic modeling for two-phase pantographic-inspired bi-continuous materials. Int J Solids Struct2020; 188–189: 282–308.
39.
AlibertJJSeppecherPdell’IsolaF.Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids2003; 8(1): 51–73.
40.
BarchiesiEEugsterSRPlacidiL, et al. Pantographic beam: A complete second gradient 1D-continuum in plane. Z angew Math Phys2019; 70(5): 135.
41.
PlacidiLdell’IsolaFBarchiesiE.Heuristic homogenization of Euler and pantographic beams. In Mechanics of Fibrous Materials and Applications. Springer, 2020, pp. 123–155.
42.
BarchiesiEKhakaloS.Variational asymptotic homogenization of beam-like square lattice structures. Math Mech Solids2019; 24(10): 3295–3318.
43.
BerezovskiAYildizdagMEScerratoD.On the wave dispersion in microstructured solids. Continuum Mech Thermodyn2018; in press.
44.
LaudatoMBarchiesiE.Non-linear dynamics of pantographic fabrics: Modelling and numerical study. In Wave Dynamics, Mechanics and Physics of Microstructured Metamaterials. Berlin: Springer, 2019, pp. 241–254.
45.
NejadsadeghiNDe AngeloMDrobnickiR, et al. Parametric experimentation on pantographic unit cells reveals local extremum configuration. Exp Mech2019; in press.
46.
NiiranenJBalobanovVKiendlJ, et al. Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro-and nano-beam models. Math Mech Solids2019; 24(1): 312–335.
47.
SolyaevYLurieSBarchiesiE, et al. On the dependence of standard and gradient elastic material constants on a field of defects. Math Mech Solids2020; 25(1): 35–45.
48.
AlessandroniSdell’IsolaFPorfiriM.A revival of electric analogs for vibrating mechanical systems aimed to their efficient control by PZT actuators. Int J Solids Struct2002; 39(20): 5295–5324.
dell’IsolaFPorfiriMVidoliS.Piezo-ElectroMechanical (PEM) structures: Passive vibration control using distributed piezoelectric transducers. C R Mecanique2003; 331(1): 69–76.
51.
LossouarnBDeüJFAucejoM.Multimodal vibration damping of a beam with a periodic array of piezoelectric patches connected to a passive electrical network. Smart Mater Struct2015; 24(11): 115037.
52.
LossouarnBDeüJFAucejoM, et al. Multimodal vibration damping of a plate by piezoelectric coupling to its analogous electrical network. Smart Mater Struct2016; 25(11): 115042.
53.
CazzaniAMalagMTurcoE.Isogeometric analysis of plane-curved beams. Math Mech Solids2014; 21(5): 562–577.
54.
CazzaniAMalagMTurcoE, et al. Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math Mech Solids2015; 21(2): 182–209.
55.
GrecoLCuomoM.B-spline interpolation of Kirchhoff–Love space rods. Computer Methods in Applied Mechanics and Engineering2013; 256: 251 – 269.
56.
GrecoLCuomoM.An implicit G1 multi patch B-spline interpolation for Kirchhoff–Love space rod. Comput Meth Appl Mech Eng2014; 269: 173–197.
57.
GrecoLCuomoM.An isogeometric implicit G1 mixed finite element for Kirchhoff space rods. Comput Meth Appl Mech Eng2016; 298: 325–349.
58.
GrecoLCuomoMContrafattoL, et al. An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Comput Meth Appl Mech Eng2017; 324: 476–511.
YaghoubiSTBalobanovVMousaviSM, et al. Variational formulations and isogeometric analysis for the dynamics of anisotropic gradient-elastic Euler–Bernoulli and shear-deformable beams. Eur J Mech A/Solids2018; 69: 113–123.
61.
CazzaniAStochinoFTurcoE.An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. Z angew Math Mech2016; 96(10): 1220–1244.
62.
BalobanovVNiiranenJ.Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity. Comput Meth Appl Mech Eng2018; 339: 137–159.
63.
AltenbachHBrsanMEremeyevVA.On a thermodynamic theory of rods with two temperature fields. Acta Mechanica2012; 223(8): 1583–1596.
64.
AltenbachHBrsanMEremeyevVA.Cosserat-type rods. In Generalized Continua from the Theory to Engineering Applications. New York: Springer, 2013, pp. 179–248.
65.
BattistiUBerchioEFerreroA, et al. Energy transfer between modes in a nonlinear beam equation. J Math Pures Appl2017; 108(6): 885–917.
66.
ChróścielewskiJSchmidtREremeyevVA.Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches. Continuum Mech Thermodyn2019; 31(1): 147–188.
67.
DeüJFGalucioACOhayonR.Dynamic responses of flexible-link mechanisms with passive/active damping treatment. Comput Struct2008; 86(3–5): 258–265.
68.
LazarusAThomasODeüJF.Finite element reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS. Finite Elements Anal Des2012; 49(1): 35–51.
69.
ThomasOSénéchalADeüJF.Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams. Nonlin Dynam2016; 86(2): 1293–1318.
70.
BoyerFGlandaisNKhalilW.Flexible multibody dynamics based on a non-linear Euler–Bernoulli kinematics. Int J Numer Meth Eng2002; 54(1): 27–59.
71.
BoyerFPrimaultD.Finite element of slender beams in finite transformations: A geometrically exact approach. Int J Numer Meth Eng2004; 59(5): 669–702.
72.
LuongoAD’AnnibaleF.Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads. Int J Non-Lin Mech2013; 55: 128–139.
73.
LuongoAD’AnnibaleF.Nonlinear hysteretic damping effects on the post-critical behaviour of the visco-elastic Beck’s beam. Math Mech Solids2017; 22(6): 1347–1365.
74.
LuongoAFerrettiMD’AnnibaleF.Paradoxes in dynamic stability of mechanical systems: Investigating the causes and detecting the nonlinear behaviors. SpringerPlus2016; 5(1): 60.
75.
LuongoAFerrettiMSeyranianA.Effects of damping on the stability of the compressed Nicolai beam. Math Mech Complex Syst2015; 3(1): 1–26.
76.
SpagnuoloMAndreausU.A targeted review on large deformations of planar elastic beams: Extensibility, distributed loads, buckling and post-buckling. Math Mech Solids2019; 24(1): 258–280.
