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Dynamic frictional contact of a viscoelastic beam

Published online by Cambridge University Press:  21 June 2006

Marco Campo
Affiliation:
Departamento de Matemática Aplicada, Facultade de Matemáticas, Campus Sur s/n, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. [email protected]; [email protected]; [email protected]
José R. Fernández
Affiliation:
Departamento de Matemática Aplicada, Facultade de Matemáticas, Campus Sur s/n, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. [email protected]; [email protected]; [email protected]
Georgios E. Stavroulakis
Affiliation:
Department of Production Engineering and Management, Technical University of Crete, 73100 Chania, Greece. [email protected]
Juan M. Viaño
Affiliation:
Departamento de Matemática Aplicada, Facultade de Matemáticas, Campus Sur s/n, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. [email protected]; [email protected]; [email protected]
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Abstract

In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformableobstacle. The beam is assumed to be situated horizontally and to move, in both horizontal andtangential directions, by the effect of applied forces. The left end of the beam is clampedand the right one is free. Its horizontal displacement is constrained because of the presenceof 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 ofthe free end, by using Tresca's law or Coulomb's law. In both cases, the variationalformulation leads to a nonlinear variational equation for the horizontal displacement coupledwith a nonlinear variational inequality for the vertical displacement. We recall an existenceand uniqueness result. Then, by using the finite element method to approximate the spatial variableand 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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2006

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References

Andrews, K.T., Shillor, M. and Wright, S., On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle. J. Elasticity 42 (1996) 130. CrossRef
Andrews, K.T., Chapman, L., Fernández, J.R., Fisackerly, M., Shillor, M., Vanerian, L. and VanHouten, T., A membrane in adhesive contact. SIAM J. Appl. Math. 64 (2003) 152169.
Andrews, K.T., Fernández, J.R. and Shillor, M., A thermoviscoelastic beam with a tip body. Comput. Mech. 33 (2004) 225234. CrossRef
Andrews, K.T., Fernández, J.R. and Shillor, M., Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod. IMA J. Appl. Math. 70 (2005) 768795. CrossRef
Bermúdez, A. and Moreno, C., Duality methods for solving variational inequalities. Comput. Math. Appl. 7 (1981) 4358. CrossRef
Campo, M., Fernández, J.R. and Viaño, J.M., Numerical analysis and simulations of a quasistatic frictional contact problem with damage. J. Comput. Appl. Math. 192 (2006) 3039. CrossRef
Chau, O., Fernández, J.R., Han, W. and Sofonea, M., A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage. Comput. Methods Appl. Mech. Eng. 191 (2002) 50075026. CrossRef
Cheng, X. and Han, W., Inexact Uzawa algorithms for variational inequalities of the second kind. Comput. Methods Appl. Mech. Eng. 192 (2003) 14511462. CrossRef
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).
J.R. Fernández, M. Shillor and M. Sofonea, Numerical analysis and simulations of quasistatic frictional wear of a beam (submitted).
Galucio, A.C., Deü, J.-F. and Ohayon, R., Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Comput. Mech. 33 (2004) 282291. CrossRef
R. Glowinski, Numerical methods for nonlinear variational problems. Springer, New York (1984).
W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity. American Mathematical Society-Intl. Press (2002).
Han, W., Kuttler, K.L., Shillor, M. and Sofonea, M., Elastic beam in adhesive contact. Int. J. Solids Struct. 39 (2002) 11451164. CrossRef
Klarbring, A., Mikelić, A. and Shillor, M., Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26 (1988) 811832. CrossRef
Kuttler, K.L., Park, A., Shillor, M. and Zhang, W., Unilateral dynamic contact of two beams. Math. Comput. Model. 34 (2001) 365384. CrossRef
T.A. Laursen, Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer, Berlin (2002).
P.D. Panagiotopoulos, Inequality problems in mechanics and applications. Convex and nonconvex energy functions. Birkhäuser Boston, Boston (1985).
Romero, I., The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput. Mech. 34 (2004) 121133. CrossRef
Romero, I. and Armero, F., 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) 16831716. CrossRef
Sofonea, M., Shillor, M. and Touzani, R., Quasistatic frictional contact and wear of a beam. Dyn. Contin. Discrete I. 8 (2000) 201218.
Stavroulakis, G.E. and Antes, H., Nonlinear boundary equation approach for inequality 2-D elastodynamics. Eng. Anal. Bound. Elem. 23 (1999) 487501. CrossRef
P. Wriggers, Computational contact mechanics. John Wiley and Sons Ltd (2002).
Zhang, H.W., He, S.Y., Li, X.S. and Wriggers, P., A new algorithm for numerical solution of 3D elastoplastic contact problems with orthotropic friction law. Comput. Mech. 34 (2004) 114.