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.

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

^{}; Fernández, José R.

^{}; Stavroulakis, Georgios E.

^{1}; Viaño, Juan M.

^{}

@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: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {295--310}, publisher = {EDP-Sciences}, volume = {40}, number = {2}, year = {2006}, doi = {10.1051/m2an:2006019}, mrnumber = {2241824}, zbl = {1137.74409}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an:2006019/} }

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: Modélisation mathématique et analyse numérique PY - 2006 SP - 295 EP - 310 VL - 40 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2006019/ DO - 10.1051/m2an:2006019 LA - en ID - M2AN_2006__40_2_295_0 ER -

%0 Journal Article %A Campo, Marco %A Fernández, José R. %A Stavroulakis, Georgios E. %A Viaño, Juan M. %T Dynamic frictional contact of a viscoelastic beam %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 295-310 %V 40 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2006019/ %R 10.1051/m2an:2006019 %G en %F 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: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 2, pp. 295-310. doi : 10.1051/m2an:2006019. http://archive.numdam.org/articles/10.1051/m2an:2006019/

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

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

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

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

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

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

, 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

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

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

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

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

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

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

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

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

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

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

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

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

,[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

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

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

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

, , and ,*Cited by Sources: *