Unilateral dynamic contact of two viscoelastic beams
Authors:
Alessia Berti and Maria Grazia Naso
Journal:
Quart. Appl. Math. 69 (2011), 477-507
MSC (2000):
Primary 74H40, 74M15, 35B40
DOI:
https://doi.org/10.1090/S0033-569X-2011-01237-7
Published electronically:
May 10, 2011
MathSciNet review:
2850742
Full-text PDF Free Access
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Additional Information
Abstract: This work is focused on a dynamic unilateral contact problem between two viscoelastic beams. Global-in-time existence of weak solutions describing the dynamics of the system is established. In addition, asymptotic longtime behavior of weak solutions is discussed: it is shown that the energy solutions decay exponentially to zero under suitable decay properties of the memory kernels.
References
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References
- K.T. Andrews, M. Shillor, and S. Wright, On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle, J. Elasticity 42 (1996), no. 1, 1–30. MR 1390198 (97h:73073)
- H. Antes and P. D. Panagiotopoulos, The boundary integral approach to static and dynamic contact problems, International Series of Numerical Mathematics, vol. 108, Birkhäuser Verlag, Basel, 1992, Equality and inequality methods. MR 1180519 (94b:73048)
- G. Bonfanti, M. Fabrizio, J.E. Muñoz Rivera, and M.G. Naso, On the energy decay for a thermoelastic contact problem involving heat transfer, J. Thermal Stresses, in press.
- G. Bonfanti, J.E. Muñoz Rivera, and M.G. Naso, Global existence and exponential stability for a contact problem between two thermoelastic beams, J. Math. Anal. Appl. 345 (2008), no. 1, 186–202. MR 2422644 (2009g:74107)
- G. Bonfanti and M.G. Naso, A dynamic contact problem between two thermoelastic beams, Applied and Industrial Mathematics in Italy III, Ser. Adv. Math. Appl. Sci., vol. 82, World Sci. Publ., Hackensack, NJ, 2009, pp. 123–132.
- M. I. M. Copetti and D. A. French, Numerical approximation and error control for a thermoelastic contact problem, Appl. Numer. Math. 55 (2005), no. 4, 439–457. MR 2176623 (2006e:74034)
- M.I.M. Copetti, Finite element approximation to a quasi-static thermoelastic problem to the contact of two rods, Appl. Numer. Math. 44 (2003), no. 1-2, 31–47. MR 1951286 (2004b:65146)
- ---, Numerical approximation of dynamic deformations of a thermoviscoelastic rod against an elastic obstacle, M2AN Math. Model. Numer. Anal. 38 (2004), no. 4, 691–706. MR 2087730 (2005e:74030)
- C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297–308. MR 0281400 (43:7117)
- G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin, 1976. MR 0521262 (58:25191)
- C. Eck, J. Jarušek, and M. Krbec, Unilateral contact problems, Variational methods and existence theorems. Pure and Applied Mathematics (Boca Raton), vol. 270, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2128865 (2006f:74049)
- C. M. Elliott and Q. Tang, A dynamic contact problem in thermoelasticity, Nonlinear Anal. 23 (1994), no. 7, 883–898. MR 1302150 (95i:73013)
- M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rational Mech. Anal. 116 (1991), no. 2, 139–152. MR 1143437 (92k:73040)
- M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153021 (93a:73034)
- M. Frémond, Non-smooth thermomechanics, Springer-Verlag, Berlin, 2002. MR 1885252 (2003g:74004)
- H. Gao and J.E. Muñoz Rivera, Global existence and decay for the semilinear thermoelastic contact problem, J. Differential Equations 186 (2002), no. 1, 52–68. MR 1941092 (2004a:74010)
- C. Giorgi and B. Lazzari, On the stability for linear viscoelastic solids, Quart. Appl. Math. 55 (1997), no. 4, 659–675. MR 1486541 (98h:73050)
- W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, AMS/IP Studies in Advanced Mathematics, vol. 30, American Mathematical Society, Providence, RI, 2002. MR 1935666 (2003m:74086)
- N. Kikuchi and J. T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 961258 (89j:73097)
- J.U. Kim, A one-dimensional dynamic contact problem in linear viscoelasticity, Math. Methods Appl. Sci. 13 (1990), no. 1, 55–79. MR 1060224 (91f:35182)
- K. L. Kuttler, A. Park, M. Shillor, and W. Zhang, Unilateral dynamic contact of two beams, Math. Comput. Modelling 34 (2001), no. 3-4, 365–384. MR 1835833 (2002d:74056)
- K.L. Kuttler and M. Shillor, Vibrations of a beam between two stops, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 8 (2001), no. 1, 93–110. MR 1824287 (2003a:74023)
- J. Lagnese and J.-L. Lions, Modelling analysis and control of thin plates, Recherches en Mathématiques Appliquées, vol. 6, Masson, Paris, 1988. MR 953313 (89k:73001)
- Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math. 54 (1996), no. 1, 21–31. MR 1373836 (97f:73019)
- J.E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math. 52 (1994), no. 4, 628–648. MR 1306041 (95j:73052)
- J.E. Muñoz Rivera and D. Andrade, Existence and exponential decay for contact problems in thermoelasticity, Appl. Anal. 72 (1999), no. 3-4, 253–273. MR 1709063 (2000d:74027)
- J.E. Muñoz Rivera and M. de Lacerda Oliveira, Exponential stability for a contact problem in thermoelasticity, IMA J. Appl. Math. 58 (1997), no. 1, 71–82. MR 1446623 (98i:73052)
- J.E. Muñoz Rivera and S. Jiang, The thermoelastic and viscoelastic contact of two rods, J. Math. Anal. Appl. 217 (1998), no. 2, 423–458. MR 1492098 (98m:73076)
- J.E. Muñoz Rivera, M.G. Naso, and F.M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl. 286 (2003), no. 2, 692–704. MR 2008858 (2004j:35205)
- J.E. Muñoz Rivera and H. Portillo Oquendo, Existence and decay to contact problems for thermoviscoelastic plates, J. Math. Anal. Appl. 233 (1999), no. 1, 56–76. MR 1684372 (2000b:74059)
- ---, Exponential decay for a contact problem with local damping, Funkcial. Ekvac. 42 (1999), no. 3, 371–387. MR 1745310 (2001h:35113)
- J.E. Muñoz Rivera and H. Portillo Oquendo, Exponential stability to a contact problem of partially viscoelastic materials, J. Elasticity 63 (2001), no. 2, 87–111. MR 1886854 (2002m:35014)
- J.E. Muñoz Rivera and J.B. Sobrinho, Existence and uniform rates of decay for contact problems in viscoelasticity, Appl. Anal. 67 (1997), no. 3-4, 175–199. MR 1614041 (98m:73050)
- M. Nakao and J.E. Muñoz Rivera, The contact problem in thermoviscoelastic materials, J. Math. Anal. Appl. 264 (2001), no. 2, 522–545. MR 1876748 (2002j:74060)
- V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math. 64 (2006), no. 3, 499–513. MR 2259051 (2007h:35211)
- F.G. Pfeiffer, Applications of unilateral multibody dynamics, Phil. Trans. R. Soc. Lond. A 359 (2001), no. 1789, 2609–2628. MR 1884318 (2002m:70016)
- M. Renardy, W.J. Hrusa, and J.A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow, 1987. MR 919738 (89b:35134)
- M. Rochdi and M. Shillor, Existence and uniqueness for a quasistatic frictional bilateral contact problem in thermoviscoelasticity, Quart. Appl. Math. 58 (2000), no. 3, 543–560. MR 1770654 (2001b:74040)
- Á. Rodríguez-Arós, J.M. Viaño, and M. Sofonea, Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory, Numer. Math. 108 (2007), no. 2, 327–358. MR 2358007 (2008m:74114)
- P. Shi and M. Shillor, Uniqueness and stability of the solution to a thermoelastic contact problem, European J. Appl. Math. 1 (1990), no. 4, 371–387. MR 1117357 (92f:73010)
- M. Shillor, M. Sofonea, and J. J. Telega, Quasistatic viscoelastic contact with friction and wear diffusion, Quart. Appl. Math. 62 (2004), no. 2, 379–399. MR 2054605 (2005b:74093)
- M. Shillor, M. Sofonea, and J.J. Telega, Models and analysis of quasistatic contact, Lect. Notes Phys., vol. 655, Springer, 2004.
- J. Simon, Compact sets in the space ${L}^p(0,{T};{B})$, Ann. Mat. Pura Appl. (4) 146 (1987), no. 1, 65–96. MR 916688 (89c:46055)
- M. Sofonea, W. Han, and M. Shillor, Analysis and approximation of contact problems with adhesion or damage, Pure and Applied Mathematics (Boca Raton), vol. 276, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2183435 (2007f:74077)
- M.E. Stavroulaki and G.E. Stavroulakis, Unilateral contact applications using FEM software, Int. J. Appl. Math. Comput. Sci. 12 (2002), no. 1, 115–125. MR 1905999 (2003c:74066)
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Additional Information
Alessia Berti
Affiliation:
Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italia
Email:
alessia.berti@ing.unibs.it
Maria Grazia Naso
Affiliation:
Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italia
Email:
naso@ing.unibs.it
Keywords:
Viscoelastic beam,
Signorini condition,
contact,
asymptotic behavior
Received by editor(s):
December 4, 2009
Published electronically:
May 10, 2011
Article copyright:
© Copyright 2011
Brown University