Analysis of a contact problem for a viscoelastic Bresse system

In this paper, we consider a contact problem between a viscoelastic Bresse beam and a deformable obstacle. The well-known normal compliance contact condition is used to model the contact. The existence of a unique solution to the continuous problem is proved using the Faedo-Galerkin method. An exponential decay property is also obtained defining an adequate Liapunov function. Then, using the finite element method and the implicit Euler scheme, a finite element approximation is introduced. A discrete stability property and a priori error estimates are proved. Finally, some numerical experiments are performed to demonstrate the decay of the discrete energy and the numerical convergence.

[1]  José R. Fernández,et al.  Dynamic vibrations of a damageable viscoelastic beam in contact with two stops , 2013 .

[2]  M. G. Naso,et al.  Unilateral dynamic contact of two viscoelastic beams , 2011 .

[3]  Stability and optimality of decay rate for a weakly dissipative Bresse system , 2015 .

[4]  M. Copetti,et al.  On the theoretical and numerical stability of the thermoviscoelastic Bresse system , 2019, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik.

[5]  Jong Uhn Kim.  A one-dimensional dynamic contact problem in linear viscoelasticity , 1990 .

[6]  Numerical approximation of dynamic deformations of a thermoviscoelastic rod against an elastic obstacle , 2004 .

[7]  Mauro de Lima Santos,et al.  Numerical Exponential Decay to Dissipative Bresse System , 2010, J. Appl. Math..

[8]  Friedrich Pfeiffer,et al.  Applications of unilateral multibody dynamics , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[9]  On the Energy Decay for a Thermoelastic Contact Problem Involving Heat Transfer , 2010 .

[10]  W. Youssef.  Contrôle et stabilisation de systèmes élastiques couplés , 2009 .

[11]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

[12]  José R. Fernández,et al.  A dynamic thermoviscoelastic contact problem with the second sound effect , 2015 .

[13]  M. Frémond,et al.  Non-Smooth Thermomechanics , 2001 .

[14]  J. M. Rivera,et al.  Stability to weak dissipative Bresse system , 2011 .

[15]  J. Oden,et al.  Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , 1987 .

[16]  Zhuangyi Liu,et al.  Energy decay rate of the thermoelastic Bresse system , 2009 .

[17]  Douglas N. Arnold,et al.  On the Range of Applicability of the Reissner–Mindlin and Kirchhoff–Love Plate Bending Models , 2002 .

[18]  T. El Arwadi,et al.  On the Stabilization of the Bresse Beam with Kelvin–Voigt Damping , 2019, Applied Mathematics & Optimization.

[19]  The Thermoelastic and Viscoelastic Contact of Two Rods , 1998 .

[20]  M. Aouadi,et al.  Analytical and numerical results for a dynamic contact problem with two stops in thermoelastic diffusion theory , 2016 .

[21]  Georges Mouret,et al.  Cours de mécanique appliquée , 1913 .

[22]  Meir Shillor,et al.  On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle , 1996 .

[23]  Anton Dammes de Pater,et al.  The Mechanics of the contact between deformable bodies : proceedings of the symposium of the International Union of Theoretical and Applied Mechanics (IUTAM), Enschede, Netherlands, 20-23 August 1974 , 1975 .

[24]  M. I. M. Copetti,et al.  Numerical approximation and error control for a thermoelastic contact problem , 2005 .

[25]  C. Bernardi,et al.  Discretization of a nonlinear dynamic thermoviscoelastic Timoshenko beam model , 2017 .

[26]  Finite element approximation to a quasi-static thermoelastic problem to the contact of two rods , 2003 .

[27]  W. Han,et al.  Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage , 2001 .

[28]  M. Nakao,et al.  The Contact Problem in Thermoviscoelastic Materials , 2001 .

[29]  M. Aouadi,et al.  A contact problem in thermoviscoelastic diffusion theory with second sound , 2017 .

[30]  G. Stavroulakis,et al.  Unilateral contact applications using FEM software , 2002 .

[31]  Michel Frémond,et al.  Contact with Adhesion , 1988 .

[32]  J. M. Rivera,et al.  Global existence and exponential stability for a contact problem between two thermoelastic beams , 2008 .

[33]  A. J. van der Merwe,et al.  Comparison of linear beam theories , 2009, Math. Comput. Model..

[34]  David J. Shippy.  The Boundary Integral Approach to Static and Dynamic Contact Problems (H. Antes and P. D. Panagiotopoulos) , 1993, SIAM Rev..

[35]  J. M. Rivera,et al.  Rates of decay to weak thermoelastic Bresse system , 2010 .

[36]  J. Lions,et al.  Inequalities in mechanics and physics , 1976 .

[37]  J. M. Muñoz Rivera,et al.  A contact problem for a thermoelastic Timoshenko beam , 2015 .

[38]  A. Wehbe,et al.  Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks , 2010 .