# Dynamic frictional contact of a viscoelastic beam

• Volume: 40, Issue: 2, page 295-310
• ISSN: 0764-583X

top

## Abstract

top
In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformable obstacle. The beam is assumed to be situated horizontally and to move, in both horizontal and tangential directions, by the effect of applied forces. The left end of the beam is clamped and the right one is free. Its horizontal displacement is constrained because of the presence of a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition. The effect of the friction is included in the vertical motion of the free end, by using Tresca's law or Coulomb's law. In both cases, the variational formulation leads to a nonlinear variational equation for the horizontal displacement coupled with a nonlinear variational inequality for the vertical displacement. We recall an existence and uniqueness result. Then, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, a numerical scheme is proposed. Error estimates on the approximative solutions are derived. Numerical results demonstrate the application of the proposed algorithm.

## How to cite

top

Campo, Marco, et al. "Dynamic frictional contact of a viscoelastic beam." ESAIM: Mathematical Modelling and Numerical Analysis 40.2 (2006): 295-310. <http://eudml.org/doc/249728>.

@article{Campo2006,
abstract = { In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformable obstacle. The beam is assumed to be situated horizontally and to move, in both horizontal and tangential directions, by the effect of applied forces. The left end of the beam is clamped and the right one is free. Its horizontal displacement is constrained because of the presence of a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition. The effect of the friction is included in the vertical motion of the free end, by using Tresca's law or Coulomb's law. In both cases, the variational formulation leads to a nonlinear variational equation for the horizontal displacement coupled with a nonlinear variational inequality for the vertical displacement. We recall an existence and uniqueness result. Then, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, a numerical scheme is proposed. Error estimates on the approximative solutions are derived. Numerical results demonstrate the application of the proposed algorithm. },
author = {Campo, Marco, Fernández, José R., Stavroulakis, Georgios E., Viaño, Juan M.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Dynamic unilateral contact; friction; viscoelastic beam; error estimates; numerical simulations.; numerical simulations},
language = {eng},
month = {6},
number = {2},
pages = {295-310},
publisher = {EDP Sciences},
title = {Dynamic frictional contact of a viscoelastic beam},
url = {http://eudml.org/doc/249728},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Campo, Marco
AU - Fernández, José R.
AU - Stavroulakis, Georgios E.
AU - Viaño, Juan M.
TI - Dynamic frictional contact of a viscoelastic beam
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/6//
PB - EDP Sciences
VL - 40
IS - 2
SP - 295
EP - 310
AB - In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformable obstacle. The beam is assumed to be situated horizontally and to move, in both horizontal and tangential directions, by the effect of applied forces. The left end of the beam is clamped and the right one is free. Its horizontal displacement is constrained because of the presence of a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition. The effect of the friction is included in the vertical motion of the free end, by using Tresca's law or Coulomb's law. In both cases, the variational formulation leads to a nonlinear variational equation for the horizontal displacement coupled with a nonlinear variational inequality for the vertical displacement. We recall an existence and uniqueness result. Then, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, a numerical scheme is proposed. Error estimates on the approximative solutions are derived. Numerical results demonstrate the application of the proposed algorithm.
LA - eng
KW - Dynamic unilateral contact; friction; viscoelastic beam; error estimates; numerical simulations.; numerical simulations
UR - http://eudml.org/doc/249728
ER -

## References

top
1. K.T. Andrews, M. Shillor and S. Wright, On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle. J. Elasticity42 (1996) 1–30.  Zbl0860.73028
2. K.T. Andrews, L. Chapman, J.R. Fernández, M. Fisackerly, M. Shillor, L. Vanerian and T. VanHouten, A membrane in adhesive contact. SIAM J. Appl. Math.64 (2003) 152–169.  Zbl1081.74034
3. K.T. Andrews, J.R. Fernández and M. Shillor, A thermoviscoelastic beam with a tip body. Comput. Mech.33 (2004) 225–234.  Zbl1067.74035
4. K.T. Andrews, J.R. Fernández and M. Shillor, Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod. IMA J. Appl. Math.70 (2005) 768–795.  Zbl1171.74386
5. A. Bermúdez and C. Moreno, Duality methods for solving variational inequalities. Comput. Math. Appl.7 (1981) 43–58.  Zbl0456.65036
6. M. Campo, J.R. Fernández and J.M. Viaño, Numerical analysis and simulations of a quasistatic frictional contact problem with damage. J. Comput. Appl. Math.192 (2006) 30–39.  Zbl1088.74037
7. O. Chau, J.R. Fernández, W. Han and M. Sofonea, A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage. Comput. Methods Appl. Mech. Eng.191 (2002) 5007–5026.  Zbl1042.74039
8. X. Cheng and W. Han, Inexact Uzawa algorithms for variational inequalities of the second kind. Comput. Methods Appl. Mech. Eng.192 (2003) 1451–1462.  Zbl1033.65045
9. P.G. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. II (1991) 17–352.
10. G. Duvaut and J.L. Lions, Inequalities in mechanics and physics. Springer-Verlag, Berlin (1976).  Zbl0331.35002
11. J.R. Fernández, M. Shillor and M. Sofonea, Numerical analysis and simulations of quasistatic frictional wear of a beam (submitted).
12. A.C. Galucio, J.-F. Deü and R. Ohayon, Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Comput. Mech.33 (2004) 282–291.  Zbl1067.74065
13. R. Glowinski, Numerical methods for nonlinear variational problems. Springer, New York (1984).  Zbl0536.65054
14. W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity. American Mathematical Society-Intl. Press (2002).  Zbl1013.74001
15. W. Han, K.L. Kuttler, M. Shillor and M. Sofonea, Elastic beam in adhesive contact. Int. J. Solids Struct.39 (2002) 1145–1164.  Zbl1012.74050
16. A. Klarbring, A. Mikelić and M. Shillor, Frictional contact problems with normal compliance. Int. J. Eng. Sci.26 (1988) 811–832.  Zbl0662.73079
17. K.L. Kuttler, A. Park, M. Shillor and W. Zhang, Unilateral dynamic contact of two beams. Math. Comput. Model.34 (2001) 365–384.  Zbl0991.74046
18. T.A. Laursen, Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer, Berlin (2002).  Zbl0996.74003
19. P.D. Panagiotopoulos, Inequality problems in mechanics and applications. Convex and nonconvex energy functions. Birkhäuser Boston, Boston (1985).  Zbl0579.73014
20. I. Romero, The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput. Mech.34 (2004) 121–133.  Zbl1138.74406
21. I. Romero and F. Armero, An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics. Int. J. Numer. Meth. Eng.54 (2002) 1683–1716.  Zbl1098.74713
22. M. Sofonea, M. Shillor and R. Touzani, Quasistatic frictional contact and wear of a beam. Dyn. Contin. Discrete I.8 (2000) 201–218.  Zbl1015.74037
23. G.E. Stavroulakis and H. Antes, Nonlinear boundary equation approach for inequality 2-D elastodynamics. Eng. Anal. Bound. Elem.23 (1999) 487–501.  Zbl0955.74076
24. P. Wriggers, Computational contact mechanics. John Wiley and Sons Ltd (2002).  Zbl1104.74002
25. H.W. Zhang, S.Y. He, X.S. Li and P. Wriggers, A new algorithm for numerical solution of 3D elastoplastic contact problems with orthotropic friction law. Comput. Mech.34 (2004) 1–14.  Zbl1072.74061

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.