Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-17T15:50:11.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Thick obstacle problems with dynamic adhesive contact

Published online by Cambridge University Press:  25 September 2008

Jeongho Ahn
Jeongho Ahn*
Affiliation:
Department of Mathematics and Statistics, Arkansas State University, P.O. Box 70, State University, AR 72467, USA. jeongho.ahn@csm.astate.edu
Get access

Abstract

In this work, we consider dynamic frictionless contact with adhesion between a viscoelastic body of the Kelvin-Voigt type and a stationary rigid obstacle, based on the Signorini's contact conditions. Including the adhesion processes modeled by the bonding field, a new version of energy function is defined. We use the energy function to derive a new form of energy balance which is supported by numerical results. Employing the time-discretization, we establish a numerical formulation and investigate the convergence of numerical trajectories. The fully discrete approximation which satisfies the complementarity conditions is computed by using the nonsmooth Newton's method with the Kanzow-Kleinmichel function. Numerical simulations of a viscoelastic beam clamped at two ends are presented.

Keywords

AdhesionSignorini's contactcomplementarity conditionstime-discretization.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 6 , November 2008 , pp. 1021 - 1045
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahn, J., A vibrating string with dynamic frictionless impact. Appl. Numer. Math. 57 (2007) 861–884. CrossRef
Ahn, J. and Stewart, D.E., Euler-Bernoulli beam with dynamic contact: Discretization, convergence, and numerical results. SIAM J. Numer. Anal. 43 (2005) 1455–1480 (electronic). CrossRef
Ahn, J. and Stewart, D.E., Existence of solutions for a class of impact problems without viscosity. SIAM J. Math. Anal. 38 (2006) 37–63 (electronic). CrossRef
Ahn, J. and Stewart, D.E., Euler-Bernoulli beam with dynamic contact: Penalty approximation and existence. Numer. Funct. Anal. Optim. 28 (2007) 1003–1026. CrossRef
J. Ahn and D.E. Stewart, Dynamic frictionless contact in linear viscoelasticity. IMA J. Numer. Anal. doi:10.1093/imanum/drm029. CrossRef
Andrews, K.T., Chapman, L., Ferández, J.R., Fisackerly, M., Shillor, M., Vanerian, L. and Vanhouten, T., A membrane in adhesive contact. SIAM J. Appl. Math. 64 (2003) 152–169.
Andrews, K.T., Kruk, S. and Shillor, M., Modelling and simulations of a bonded rod. Math. Comput. Model. 42 (2005) 553–572.
J.H. Bramble and X. Zhang, The Analysis of Multigrid Methods, Handbook of Numerical Analysis VII. North-Holland, Amsterdam (2000).
D. Candeloro and A. Volčič, Radon-Nikodým theorems, Vol. I. North Holland/Elsevier (2002).
Chau, O., Ferández, J.R., Shillor, M. and Sofonea, M., Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion. J. Comput. Appl. Math. 159 (2003) 431–465. CrossRef
Chau, O., Shillor, M. and Sofonea, M., Dynamic frictionless contact with adhesion. Z. Angew. Math. Phys. 55 (2004) 32–47. CrossRef
F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research I, II. Springer-Verlag, New York (2003).
Ferández, J.R., Shillor, M. and Sofonea, M., Analysis and numerical simulations of a dynamic contact problem with adhesion. Math. Comput. Modelling 37 (2003) 1317–1333. CrossRef
Frémond, M., Équilibre des structures qui adhèrent à leur support. C. R. Acad. Sci. Paris Sér. II 295 (1982) 913–916.
Frémond, M., Adhérence des solides. J. Méc. Théor. Appl. 6 (1987) 383–407.
M. Frémond, Contact with adhesion, in Topics Nonsmooth Mechanics, J.J. Moreau, P.D. Panagiotopoulos and G. Strang Eds. (1988) 157–186
M. Frémond, E. Sacco, N. Point and J.M. Tien, Contact with adhesion, in ESDA Proceedings of the 1996 Engineering Systems Design and Analysis Conference, A. Lagarde and M. Raous Eds., ASME, New York (1996) 151–156.
Han, W., Kuttler, K.L., Shillor, M. and Sofonea, M., Elastic beam in adhesive contact. Int. J. Solids Structures 39 (2002) 1145–1164. CrossRef
Jianu, L., Shillor, M. and Sofonea, M., A viscoelastic frictionless contact problem with adhesion. Appl. Anal. 80 (2001) 233–255. CrossRef
Kanzow, C. and Kleinmichel, H., A new class of semismooth Newton-type methods for nonlinear complementarity problems. Comput. Optim. Appl. 11 (1998) 227–251. CrossRef
K. Kuttler, Modern Analysis. CRC Press, Boca Raton, FL, USA (1998).
Lebeau, G. and Schatzman, M., A wave problem in a half-space with a unilateral contraint at the boundary. J. Diff. Eq. 53 (1984) 309–361. CrossRef
Petrov, A. and Schatzman, M., Viscoélastodynamique monodimensionnelle avec conditions de Signorini. C. R. Acad. Sci. Paris Sér. I 334 (2002) 983–988. CrossRef
Qi, L.Q. and Sun, J., A nonsmooth version of Newton's method. Math. Program. 58 (1993) 353–367. CrossRef
Raous, M., Cangémi, L. and Cocu, M., A consistent model coupling adhesion, friction, and unilateral contact. Comput. Methods Appl. Mech. Engrg. 177 (1999) 383–399. CrossRef
Schatzman, M., A hyperbolic problem of second order with unilateral constraints: the vibrating string with a concave obstacle. J. Math. Anal. Appl. 73 (1980) 138–191. CrossRef
M. Shillor, M. Sofonea and J. Telega, Models and Analysis of Quasistatic Contact, Lect. Notes Phys. 655. Springer, Berlin-Heidelberg-New York (2004).
M. Sofonea, W. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics 276. Chapman-Hall/CRC Press, New York (2006).
Stewart, D.E., Convolution complementarity problems with application to impact problems. IMA J. Appl. Math. 71 (2006) 92–119. CrossRef
Stewart, D.E., Differentiating complementarity problems and fractional index convolution complementarity problems. Houston J. Math. 33 (2007) 301–322.
D.E. Stewart, Energy balance for viscoelastic bodies in frictionless contact. (Submitted).
M.E. Taylor, Partial Differential Equations 1, Applied Mathematical Sciences 115. Springer-Verlag, New York (1996).
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam, New York (1978).
J. Wloka, Partial Differential Equations. Cambridge University Press (1987).