# Dynamic frictional contact of a viscoelastic beam

Marco Campo; José R. Fernández; Georgios E. Stavroulakis; Juan M. Viaño

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

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

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topCampo, 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- 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
- 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
- K.T. Andrews, J.R. Fernández and M. Shillor, A thermoviscoelastic beam with a tip body. Comput. Mech.33 (2004) 225–234. Zbl1067.74035
- 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
- A. Bermúdez and C. Moreno, Duality methods for solving variational inequalities. Comput. Math. Appl.7 (1981) 43–58. Zbl0456.65036
- 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
- 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
- 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
- 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.
- G. Duvaut and J.L. Lions, Inequalities in mechanics and physics. Springer-Verlag, Berlin (1976). Zbl0331.35002
- J.R. Fernández, M. Shillor and M. Sofonea, Numerical analysis and simulations of quasistatic frictional wear of a beam (submitted).
- 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
- R. Glowinski, Numerical methods for nonlinear variational problems. Springer, New York (1984). Zbl0536.65054
- W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity. American Mathematical Society-Intl. Press (2002). Zbl1013.74001
- 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
- A. Klarbring, A. Mikelić and M. Shillor, Frictional contact problems with normal compliance. Int. J. Eng. Sci.26 (1988) 811–832. Zbl0662.73079
- 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
- T.A. Laursen, Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer, Berlin (2002). Zbl0996.74003
- P.D. Panagiotopoulos, Inequality problems in mechanics and applications. Convex and nonconvex energy functions. Birkhäuser Boston, Boston (1985). Zbl0579.73014
- 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
- 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
- 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
- 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
- P. Wriggers, Computational contact mechanics. John Wiley and Sons Ltd (2002). Zbl1104.74002
- 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

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