Energy balance for viscoelastic bodies in frictionless contact
Author:
David E. Stewart
Journal:
Quart. Appl. Math. 67 (2009), 735-743
MSC (2000):
Primary 74M20; Secondary 35L85, 49J40
DOI:
https://doi.org/10.1090/S0033-569X-09-01161-8
Published electronically:
May 14, 2009
MathSciNet review:
2588233
Full-text PDF Free Access
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Abstract: In this paper it is shown that the change in the energy for a linearly viscoelastic body (with Kelvin–Voigt type viscosity) in frictionless contact with a rigid obstacle can be accounted for by viscous losses and the work done by external forces. Thus there is no change in the energy due to impacts, unlike the case of rigid-body dynamics. The result can be extended to a wide class of dynamic viscoelastic boundary thin obstacle problems of similar type.
References
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References
- Jeongho Ahn and David E. Stewart. An Euler–Bernoulli beam with dynamic contact: Discretization, convergence and numerical results. SIAM J. Numer. Anal., 43(4):1455–1480, 2005. MR 2182136 (2006e:35319)
- Jeongho Ahn and David E. Stewart. Existence of solutions for a class of impact problems without viscosity. SIAM J. Mathematical Analysis, 38(1):37–63, 2006. MR 2217307 (2007g:35158)
- Kevin 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(1):1–30, 1996. MR 1390198 (97h:73073)
- C. Baiocchi and A. Capelo. Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, Chichester, New York, 1984. MR 745619 (86e:49018)
- A. Bamberger and M. Schatzman. New results on the vibrating string with a continuous obstacle. SIAM J. Math. Anal., 14(3):560–595, 1983. MR 697529 (84g:35106)
- J. Bergh and J. Löfström. Interpolation Spaces, volume 223 of Grundlehren der mathmatischen Wissenschaften. Springer-Verlag, Berlin, Heidelberg, New York, 1976.
- Marius Cocu. Existence of solutions of Signorini problems with friction. Internat. J. Engrg. Sci., 22(5):567–575, 1984. MR 750004 (86g:73050)
- Marius Cocu and Jean-Marc Ricaud. Existence results for a class of implicit evolution inequalities and applications to dynamic unilateral contact problems with friction. C. R. Acad. Sci. Paris, Sér. I, 329:839–844, 1999. MR 1724551 (2000i:34122)
- Marius Cocu and Jean-Marc Ricaud. Analysis of a class of implicit evolution inequalities associated to viscoelastic dynamic contact problems with friction. Internat. J. Engrg. Sci., 38(14):1535–1552, 2000. MR 1763039 (2001e:74066)
- J. Diestel and J. J. Uhl, Jr. Vector Measures, volume 15 of Mathematical Surveys. Amer. Math. Soc., Providence, RI, 1977. MR 0453964 (56:12216)
- F. Facchinei and J.-S. Pang. Finite-Dimensional Variational Inequalities and Complementarity Problems I. Springer Series in Operations Research. Springer, New York, 2003.
- J. Jarušek and C. Eck. Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions. Math. Models Methods Appl. Sci., 9(1):11–34, 1999. MR 1671535 (2000b:74058)
- Jiří Jarušek and Christof Eck. Dynamic contact problems with friction in linear viscoelasticity.C. R. Acad. Sci. Paris Sér. I Math., 322(5):497–502, 1996. MR 1381792 (97f:73044)
- N. Kikuchi and J. T. Oden. Contact problems in elasticity: a study of variational inequalities and finite element methods, volume 8 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 961258 (89j:73097)
- Jong Uhn Kim. A boundary thin obstacle problem for a wave equation. Commun. Partial Differential Equations, 14(8&9):1011–1026, 1989. MR 1017060 (91a:35121)
- Kenneth Kuttler and Meir Shillor. Dynamic contact with Signorini’s condition and slip rate dependent friction. Electron. J. Differential Equations, No. 83, 21 pp. (electronic), 2004. MR 2075422 (2005d:74026)
- Kenneth L. Kuttler and Meir Shillor. Vibrations of a beam between two stops. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 8(1):93–110, 2001. MR 1824287 (2003a:74023)
- G. Lebeau and M. Schatzman. A wave problem in a half-space with a unilateral constraint at the boundary. J. Differential Equations, 53:309–361, 1984. MR 752204 (86b:35108)
- Adrien Petrov and Michelle Schatzman. Viscoélastodynamique monodimensionnelle avec conditions de Signorini. C. R. Math. Acad. Sci. Paris, 334(11):983–988, 2002. MR 1913722 (2003e:35214)
- E. J. Routh. A Treatise on the Dynamics of a System of Rigid Bodies. MacMillan, London, 1860.
- Michelle Schatzman. A hyperbolic problem of second order with unilateral constraints: the vibrating string with a concave obstacle. J. Math. Anal. Appl., 73(1):138–191, 1980. MR 560941 (81d:35047)
- Michelle Schatzman and Michel Bercovier. Numerical approximation of a wave equation with unilateral constraints. Math. Comp., 53(187):55–79, 1989. MR 969491 (89k:65112)
- David E. Stewart. Convolution complementarity problems with application to impact problems. IMA J. Applied Math., 71(1):92–119, 2006. MR 2203045 (2007f:90117)
- David E. Stewart. Differentiating complementarity problems and fractional index convolution complementarity problems. Houston J. Mathematics, 33(1):301–322, 2006. MR 2287857 (2008b:90128)
- Hans Triebel. Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, New York, 1978. MR 503903 (80i:46032b)
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Additional Information
David E. Stewart
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email:
dstewart@math.uiowa.edu
Keywords:
Impact,
viscoelasticity,
energy balance,
dynamic variational inequalities
Received by editor(s):
July 25, 2008
Published electronically:
May 14, 2009
Additional Notes:
This work was supported in part by the NSF under grant DMS-0139709.
Article copyright:
© Copyright 2009
Brown University