We present a new invariant numerical formulation suitable for the analysis of Kirchhoff–Love rod model based on the smallest rotation map for which the two kinematic descriptors are the placement of the centroid curve and the rotation angle that describes the orientation of the cross-section. In order to guarantee objectivity with respect to a rigid body motion, the spherical linear interpolation (slerp) for the rotations is introduced. A new hierarchical interpolation for the rotation angle of the cross-section is proposed, composed by the drilling rotation (with respect to the unit tangent of the centroid curve) contained in the geodetic interpolation of the ends’ rotations plus an additional polynomial correction term. This correction term is needed in order to enhance the accuracy of the proposed finite element (FE) formulation. The FE model implicitly guarantees the G1 continuity conditions, thanks to a change in the parametrization of the centroid curve that leads to a generalization of the one-dimensional (1D) Hermitian interpolation. A mixed formulation is used in order to avoid locking and improve the computational efficiency of the method. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the mixed functional for which the Levi-Civita connection of the configurations manifold of the rod is needed. The objectivity, the robustness, and accuracy of the G1-conforming formulation are confirmed by means of several numerical examples.
SimoJ. A finite strain beam formulation. The three-dimensional dynamics. Part I. Comput Methods Appl Mech Eng1985; 49(1): 55–70.
2.
SimoJVu-QuocL.A three-dimensional finite-strain rod model. Part II: computational aspects. Comput Methods Appl Mech Eng1986; 58(1): 79–116.
3.
RomeroIGebhardtC.Variational principles for nonlinear Kirchhoff rods. Acta Mechanica2020; 231: 625–647.
4.
GrecoLCuomoM.B-spline interpolation of Kirchhoff-Love space rods. Comput Methods Appl Mech Eng2013; 256: 251–269.
5.
GrecoLCuomoM.Consistent tangent operator for an exact Kirchhoff rod model. Contin Mech Thermodyn2015; 27: 861–877.
6.
GrecoLCuomoM.An implicit G1 multi patch B-spline interpolation for Kirchhoff-Love space rod. Comput Methods Appl Mech Eng2014; 269: 173–197.
7.
GrecoLCuomoM.An isogeometric implicit G1 mixed finite element for Kirchhoff space rods. Comput Methods Appl Mech Eng2016; 298: 325–349.
8.
GrecoL.An iso-parametric G1-conforming finite element for the nonlinear analysis of Kirchhoff rod. Part I: the 2D case. Contin Mech Thermodyn2020; 32: 1473–1496.
9.
GrecoLScrofaniACuomoM.A non-linear symmetric G1-conforming Bézier finite element formulation for the analysis of Kirchhoff beam assemblies. Comput Methods Appl Mech Eng2021; 387: 114176.
10.
GrecoLCuomoMCastelloD, et al. An updated lagrangian Bézier finite element formulation for the analysis of slender beams. Math Mech Solids2022; 27(10): 2110–2138.
11.
ShoemakeK.Animating rotation with quaternion curves. SIGGRAPH Comput Graph1985; 19(3): 245–254.
12.
ParkFRavaniB.Smooth invariant interpolation of rotations. ACM Trans Graph1997; 16(3): 277–295.
13.
MüllerABischoffM.A consistent finite element formulation of the geometrically non-linear Reissner-Mindlin shell model. Arch Comput Methods Eng2022; 29: 3387–3434.
CrisfieldMJelenićG.Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proc R Soc Lond1999; 455(1983): 1125–1147.
16.
RomeroI.The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput Mech2004; 34: 121–133.
17.
GhoshSRoyD.Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam. Comput Methods Appl Mech Eng2008; 198: 555–571.
18.
GhoshSRoyD.A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization. Comput Mech2009; 44: 103–119.
19.
CardonaAGeradinM.A beam finite element non-linear theory with finite rotations. Int J Numer Methods Eng1988; 26: 2403–2438.
20.
ArmeroFValverdeJ.Invariant Hermitian finite elements for thin Kirchhoff rods. I: the linear plane case. Comput Methods Appl Mech Eng2012; 213-216: 427–457.
21.
ArmeroFValverdeJ.Invariant Hermitian finite elements for thin Kirchhoff rods. II: the linear three-dimensional case. Comput Methods Appl Mech Eng2012; 213-216: 458–485.
MagisanoDLeonettiLMadeoA, et al. A large rotation finite element analysis of 3D beams by incremental rotation vector and exact strain measure with all the desirable features. Comput Methods Appl Mech Eng2020; 361: 112811.
24.
MeierCPoppAWallW.An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods. Comput Methods Appl Mech Eng2014; 278: 445–478.
25.
MeierCPoppAWallW.Geometrically exact finite element formulations for slender beams: Kirchhoff-love theory versus simo-reissner theory. Arch Comput Methods Eng2019; 26: 163–243.
26.
HarschJCapobiancoGEugsterS.Finite element formulations for constrained spatial nonlinear beam theories. Math Mech Solids2021; 26(12): 1838–1863.
27.
HarschJSailerSEugsterS.A total Lagrangian, objective and intrinsically locking-free Petrov-Galerkin SE(3) Cosserat rod finite element formulation. Int J Numer Methods Eng2023; 124(13): 2965–2994.
28.
EugsterSHarschJ.A family of total Lagrangian Petrov-Galerkin cosserat rod finite element formulations. GAMM Mitt2023; 46: e202300008.
MarinoEKiendlJDe LorenzisL.Isogeometric collocation for implicit dynamics of three-dimensional beams undergoing finite motions. Comput Methods Appl Mech Eng2019; 356: 548–570.
37.
IgnestiDFerriGAuricchioF, et al. An improved isogeometric collocation formulation for spatial multi-patch shear-deformable beams with arbitrary initial curvature. Comput Methods Appl Mech Eng2023; 403: 115722.
38.
GiorgioI.A discrete formulation of Kirchhoff rods in large-motion dynamics. Math Mech Solids2020; 25(5): 1081–1100.
39.
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): e201800121.
40.
GiorgioIDel VescovoD.Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators. Math Mech Complex Syst2019; 7(2): 159–174.
41.
GiorgioI.A variational formulation for one-dimensional linear thermo-viscoelasticity. Math Mech Complex Syst2021; 9(4): 397–412.
42.
SpagnuoloMAndreausU.A targeted rewiev on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling. Math Mech Solids2019; 24(1): 258–280.
43.
CostantineJTawkYMaquedaI, et al. UHF deployable helical antennas for CubeSats. IEEE Trans Antennas Propag2016; 64(9): 3752–3759.
44.
SchikoreJSchlingEOberbichlerT, et al. Kinetics and design of semi-compliant grid mechanisms. In: BaverelOPottmannHMuellerC, et al. (eds) Advances in architectural geometry AAG 2020. Paris: Presse de ponts and chaussée, 2021, pp. 108–129.
45.
YuTDreierLMarmoF, et al. Numerical modeling of static equilibria and bifurcations in bigons and bigon rings. Journal of the Mechanics and Physics of Solids2021; 152: 104459.
46.
CiallellaAGiorgioIEugsterS, et al. Generalized beam model for the analysis of wave propagation with a symmetric pattern of deformation in planar pantographic sheet. Wave Motion2022; 113: 102986.
47.
La ValleGSpagnuoloMTurcoE, et al. A new torsional energy for pantographic sheets. Z Angew Math Phys2023; 74(2): 67.
48.
CiallellaALa ValleGVintacheA, et al. Deformation mode in 3-point flexure on pantographic block. Int J Solids Struct2023; 265-266: 112129.
49.
La ValleGCiallellaAFalsoneG. The effect of local random defects on the response of pantographic sheets. Math Mech Solids2022; 27(10): 2147–2169.
50.
AlibertJJSeppecherPdell’IsolaF.Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids2003; 8(1): 51–73.
51.
FalsoneGLa ValleG.A homogenized theory for functionally graded euler-bernoulli and timoshenko beams. Acta Mech2019; 230: 3511–3523.
52.
De AngeloMPlacidiLNejadsadeghiN, et al. Non-standard timoshenko beam model for chiral metamaterial: identification of stiffness parameters. Mech Res Commun2020; 103: 103462.
53.
BarchiesiEdell’IsolaFBersaniA, et al. Equilibria determination of elastic articulated duoskelion beams in 2D via Riks-type algorithm. International Journal of Non-Linear Mechanics2021; 128: 103628.
54.
SpagnuoloMRecciaECiallellaA, et al. Matrix-embedded metamaterials: applications for the architectural heritage. Math Mech Solids2022; 27(10): 2275–2286.
55.
VoDBorkovićANanakornP, et al. Dynamic multi-patch isogeometric analysis of planar Euler-Bernoulli beams. Comput Methods Appl Mech Eng2020; 372: 113435.
SimoJFoxD.On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization. Comput Methods Appl Mech Eng1989; 72(3): 267–304.
58.
RebelG.Finite rotation shell theory including drill rotations and its finite element implementation. PhD Thesis, Delft University Press, Delft, 1998.
59.
MäkinenJ.Rotation manifold SO(3) and its tangential vectors. Comput Mech2008; 42(6): 907–919.
60.
HallB.Lie groups, Lie algebras, and representations. New York: Springer, 2003.
61.
IbrahimbegovićAFreyFKožarI.Computational aspects of vector-like parametrization of three-dimensional finite rotations. Int J Numer Methods Eng1995; 38(21): 3653–3673.
CrisfieldM.Non-linear finite element analysis of solids and structures. Vol. 2: advanced topics. Chichester: John Wiley & Sons, 1997.
64.
HansonA.Visualizing quaternions. The Morgan Kaufmann series in interactive 3D technology. Burlington, MA: Morgan Kaufmann, 2006.
65.
JelenićGCrisfieldM.Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput Methods Appl Mech Eng1999; 171: 141–171.
66.
IbrahimbegovićATaylorR.On the role of frame invariance in structural mechanics models at finite rotations. Comput Methods Appl Mech Eng2002; 191: 5159–5176.
67.
BatheKBolourchiS.Large displacement analysis of three-dimensional beam structures. Int J Numer Methods Eng1979; 14(7): 961–986.
68.
LoSH.Geometrically nonlinear formulation of 3D finite strain beam element with large rotations. Comput Struct1992; 44(1-2): 147–157.
J KapaniaRLij. A formulation and implementation of geometrically exact curved beam elements incorporating finite strains and finite rotations. Comput Mech2003; 30: 444–459.
76.
GotoYWatanabeYKasugaiT, et al. Elastic buckling phenomenon applicable to deployable rings. Int J Solids Struct1992; 29(7): 893–909.
77.
da Costa e SilvaCMaassenSPimentaP, et al. A simple finite element for the geometrically exact analysis of Bernoulli-Euler rods. Comput Mech2020; 65: 905–923.
