[en] This paper presents a test case to help validate simulation codes for contact problems involving beams. A closed form solution is derived and the comparison is made with a finite element (FE) implementation that uses the mortar method for enforcing the contact constraints. The test case consists of a semi-infinite cantilever beam subjected to a constant distributed load and experiencing frictionless contact with a straight rigid substrate. Both an Euler-Bernoulli and a Timoshenko beam model are considered and the influence of the differing kinematic hypotheses is analyzed. In the case of the Euler-Bernoulli beam the distributed contact force is equal to the load along the contact region except at the boundary where a point load appears. On the contrary, the rigid substrate exerts a fully distributed load on the Timoshenko beam which decays exponentially from the first contact point and tends towards the applied load. The rate of decay depends on the relative shear deformability. Moreover, whereas in the first case the transverse shear force is discontinuous, it becomes continuous when allowing for shear deformation. An example of benchmarking is given for a particular FE code. The error with respect to the exact solution can be computed and it is shown that the numerical solution converges to the analytic solution when the FE mesh is refined.
Disciplines :
Engineering, computing & technology: Multidisciplinary, general & others
Author, co-author :
Bosten, Armin ; Université de Liège - ULiège > Aérospatiale et Mécanique (A&M) ; Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany
Denoël, Vincent ; Université de Liège - ULiège > Département ArGEnCo > Analyse sous actions aléatoires en génie civil
Cosimo, Alejandro ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Linn, Joachim; Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany
Bruls, Olivier ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Language :
English
Title :
A beam contact benchmark with analytic solution
Publication date :
2023
Journal title :
ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik
This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska‐Curie grant agreement No 860124. The present paper only reflects the author's view. The European Commission and its Research Executive Agency (REA) are not responsible for any use that may be made of the information it contains.
Arnold, M., Brüls, O., Linn, J.: THREAD-numerical modeling of highly flexible structures for industrial applications. In: Celledoni, E., Münch, A. (eds.) Mathematics with Industry: Driving Innovation-ECMI Annual Report 20A9, pp. 20–25 (2019)
Meier, C., Grill, M.J., Wall, W.A., Popp, A.: Geometrically exact beam elements and smooth contact schemes for the modeling of fiber-based materials and structures. Int. J. Solids Struct. 154, 124–146 (2018)
Durville, D.: Contact-friction modeling within elastic beam assemblies: an application to knot tightening. Comput. Mech. 49, 687–707 (2012)
Daviet, G., Bertails-Descoubes, F., Boissieux, L.: A hybrid iterative solver for robustly capturing coulomb friction in hair dynamics. ACM T. Graphic. Proc. of SIGGRAPH Asia. 30, 1–12 (2011)
Konyukhov, A., Shala, S.: New benchmark problems for verification of the curve-to-surface contact algorithm based on the generalized Euler-Eytelwein problem. Numer. Meth. Engrg. 123, 411–443 (2022)
Bertails, F., Audoly, B., Cani, M.-P., Querleux, B., Leroy, F., Lévêque, J.-L.: Super-helices for predicting the dynamics of natural hair. ACM Trans. Graph. 25, 1180–1187 (2006)
Meier, C., Popp, A., Wall, W.A.: An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods. Comput. Methods Appl. Mech. Eng. 278, 445–478 (2014)
Sonneville, V., Cardona, A., Brüls, O.: Geometrically exact beam finite element formulated on the special Euclidean group SE (3). Comput. Methods Appl. Mech. Engrg. 268, 451–474 (2014)
Cottenceau, E., Thomas, O., Véron, P., Alochet, M., Deligny, R.: Finite Elem. Anal. Dis. 139, 14 (2018)
Lang, H., Linn, J., Arnold M.,: Multi-body dynamics simulation of geometrically exact Cosserat rods. Multibody Syst. Dyn. 25, 285–312 (2011)
Géradin, M., Rixen, D.: Mechanical Vibrations: Theory and Application to Structural Dynamics. Wiley and Sons, Chichester, UK (2015)
Essenburg, F.: On the significance of the inclusion of the effect of transverse normal strain in problems involving beams with surface constraints. ASME J. Appl. Mech. 42, 127–132 (1975)
Naghdi, P.M., Rubin, M.B.: On the significance of normal cross-sectional extension in beam theory with application to contact problems. Int. J. Solids Struct. 25, 249–265 (1989)
Soong, T.C., Choi, I.: An elastica that involves continuous and multiple discrete contacts with a boundary. Int. J. Mech. Scs. 28, 1–10 (1986)
Denoël, V., Detournay, E.: Eulerian formulation of constrained elastica. Int. J. Sol. Struct. 48, 625–636 (2011)
Chen, J.-S.: On the contact behavior of a buckled Timoshenko beam constrained laterally by a plane wall. Acta Mech. 222, 225–232 (2011)
Denoël, V.: Advantages of a semi-analytical approach for the analysis of an evolving structure with contacts. Commun. Numer. Meth. Engng. 24, 1667-1683 (2008)
Huynen, A., Detournay, E., Denoël, V.: Eulerian formulation of elastic rods. Proc. R. Soc. A 472 (2016)
Hinch, E.J.: Perturbation Methods, p. 160. Cambridge University Press, (1991)
Sonneville, V., Brüls, O., Bauchau, O.: Interpolation schemes for geometrically exact beams: a motion approach. Int. J. Numer. Methods Eng. 112, 1129–1153 (2017)
Bosten, A., Cosimo, A., Linn, J., Brüls, O.: A mortar formulation for frictionless line-to-line beam contact. Multibody Syst. Dyn. 54, 31–52 (2021)
Tomec, J., Jelenić, G.: Analysis of static frictionless beam-to-beam contact using mortar method. Multibody Syst. Dyn. 55, 293–322 (2022)
Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Eng. 49, 55–70 (1985)
