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.

Classification: 65N15, 65N30, 74D05, 74M10, 74M15, 74S05, 74S20

Keywords: dynamic unilateral contact, friction, viscoelastic beam, error estimates, numerical simulations

@article{M2AN_2006__40_2_295_0, author = {Campo, Marco and Fern\'andez, Jos\'e R. and Stavroulakis, Georgios E. and Via\~no, Juan M.}, title = {Dynamic frictional contact of a viscoelastic beam}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {40}, number = {2}, year = {2006}, pages = {295-310}, doi = {10.1051/m2an:2006019}, zbl = {1137.74409}, mrnumber = {2241824}, language = {en}, url = {http://www.numdam.org/item/M2AN_2006__40_2_295_0} }

Campo, Marco; Fernández, José R.; Stavroulakis, Georgios E.; Viaño, Juan M. Dynamic frictional contact of a viscoelastic beam. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, pp. 295-310. doi : 10.1051/m2an:2006019. http://www.numdam.org/item/M2AN_2006__40_2_295_0/

[1] On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle. J. Elasticity 42 (1996) 1-30. | Zbl 0860.73028

, and ,[2] A membrane in adhesive contact. SIAM J. Appl. Math. 64 (2003) 152-169. | Zbl 1081.74034

, , , , , and ,[3] A thermoviscoelastic beam with a tip body. Comput. Mech. 33 (2004) 225-234. | Zbl 1067.74035

, and ,[4] Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod. IMA J. Appl. Math. 70 (2005) 768-795. | Zbl pre05016022

, and ,[5] Duality methods for solving variational inequalities. Comput. Math. Appl. 7 (1981) 43-58. | Zbl 0456.65036

and ,[6] Numerical analysis and simulations of a quasistatic frictional contact problem with damage. J. Comput. Appl. Math. 192 (2006) 30-39. | Zbl 1088.74037

, and ,[7] A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage. Comput. Methods Appl. Mech. Eng. 191 (2002) 5007-5026. | Zbl 1042.74039

, , and ,[8] Inexact Uzawa algorithms for variational inequalities of the second kind. Comput. Methods Appl. Mech. Eng. 192 (2003) 1451-1462. | Zbl 1033.65045

and ,[9] 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. | Zbl 0875.65086

,[10] Inequalities in mechanics and physics. Springer-Verlag, Berlin (1976). | MR 521262 | Zbl 0331.35002

and ,[11] Numerical analysis and simulations of quasistatic frictional wear of a beam (submitted).

, and ,[12] Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Comput. Mech. 33 (2004) 282-291. | Zbl 1067.74065

, and ,[13] Numerical methods for nonlinear variational problems. Springer, New York (1984). | MR 737005 | Zbl 0536.65054

,[14] Quasistatic contact problems in viscoelasticity and viscoplasticity. American Mathematical Society-Intl. Press (2002). | MR 1935666 | Zbl 1013.74001

and ,[15] Elastic beam in adhesive contact. Int. J. Solids Struct. 39 (2002) 1145-1164. | Zbl 1012.74050

, , and ,[16] Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26 (1988) 811-832. | Zbl 0662.73079

, and ,[17] Unilateral dynamic contact of two beams. Math. Comput. Model. 34 (2001) 365-384. | Zbl 0991.74046

, , and ,[18] Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer, Berlin (2002). | MR 1902698 | Zbl 0996.74003

,[19] Inequality problems in mechanics and applications. Convex and nonconvex energy functions. Birkhäuser Boston, Boston (1985). | MR 896909 | Zbl 0579.73014

,[20] The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput. Mech. 34 (2004) 121-133. | Zbl 1138.74406

,[21] 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. | Zbl 1098.74713

and ,[22] Quasistatic frictional contact and wear of a beam. Dyn. Contin. Discrete I. 8 (2000) 201-218. | Zbl 1015.74037

, and ,[23] Nonlinear boundary equation approach for inequality 2-D elastodynamics. Eng. Anal. Bound. Elem. 23 (1999) 487-501. | Zbl 0955.74076

and ,[24] Computational contact mechanics. John Wiley and Sons Ltd (2002).

,[25] A new algorithm for numerical solution of 3D elastoplastic contact problems with orthotropic friction law. Comput. Mech. 34 (2004) 1-14. | Zbl 1072.74061

, , and ,