Thick obstacle problems with dynamic adhesive contact

Ahn, Jeongho

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 6, pp. 1021-1045.

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.

[1] J. Ahn, A vibrating string with dynamic frictionless impact. Appl. Numer. Math. 57 (2007) 861-884. | MR | Zbl

[2] J. Ahn and D.E. Stewart, Euler-Bernoulli beam with dynamic contact: Discretization, convergence, and numerical results. SIAM J. Numer. Anal. 43 (2005) 1455-1480 (electronic). | MR | Zbl

[3] J. Ahn and D.E. Stewart, Existence of solutions for a class of impact problems without viscosity. SIAM J. Math. Anal. 38 (2006) 37-63 (electronic). | MR | Zbl

[4] J. Ahn and D.E. Stewart, Euler-Bernoulli beam with dynamic contact: Penalty approximation and existence. Numer. Funct. Anal. Optim. 28 (2007) 1003-1026. | MR | Zbl

[5] J. Ahn and D.E. Stewart, Dynamic frictionless contact in linear viscoelasticity. IMA J. Numer. Anal. doi:10.1093/imanum/drm029. | MR | Zbl

[6] K.T. Andrews, L. Chapman, J.R. Ferández, M. Fisackerly, M. Shillor, L. Vanerian and T. Vanhouten, A membrane in adhesive contact. SIAM J. Appl. Math. 64 (2003) 152-169. | MR | Zbl

[7] K.T. Andrews, S. Kruk and M. Shillor, Modelling and simulations of a bonded rod. Math. Comput. Model. 42 (2005) 553-572. | MR | Zbl

[8] J.H. Bramble and X. Zhang, The Analysis of Multigrid Methods, Handbook of Numerical Analysis VII. North-Holland, Amsterdam (2000). | MR | Zbl

[9] D. Candeloro and A. Volčič, Radon-Nikodým theorems, Vol. I. North Holland/Elsevier (2002). | MR | Zbl

[10] O. Chau, J.R. Ferández, M. Shillor and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion. J. Comput. Appl. Math. 159 (2003) 431-465. | MR | Zbl

[11] O. Chau, M. Shillor and M. Sofonea, Dynamic frictionless contact with adhesion. Z. Angew. Math. Phys. 55 (2004) 32-47. | MR | Zbl

[12] 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). | Zbl

[13] J.R. Ferández, M. Shillor and M. Sofonea, Analysis and numerical simulations of a dynamic contact problem with adhesion. Math. Comput. Modelling 37 (2003) 1317-1333. | MR

[14] M. Frémond, Équilibre des structures qui adhèrent à leur support. C. R. Acad. Sci. Paris Sér. II 295 (1982) 913-916. | MR | Zbl

[15] M. Frémond, Adhérence des solides. J. Méc. Théor. Appl. 6 (1987) 383-407. | Zbl

[16] M. Frémond, Contact with adhesion, in Topics Nonsmooth Mechanics, J.J. Moreau, P.D. Panagiotopoulos and G. Strang Eds. (1988) 157-186 | MR | Zbl

[17] 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.

[18] W. Han, K.L. Kuttler, M. Shillor and M. Sofonea, Elastic beam in adhesive contact. Int. J. Solids Structures 39 (2002) 1145-1164. | MR | Zbl

[19] L. Jianu, M. Shillor and M. Sofonea, A viscoelastic frictionless contact problem with adhesion. Appl. Anal. 80 (2001) 233-255. | MR | Zbl

[20] C. Kanzow and H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems. Comput. Optim. Appl. 11 (1998) 227-251. | MR | Zbl

[21] K. Kuttler, Modern Analysis. CRC Press, Boca Raton, FL, USA (1998). | Zbl

[22] G. Lebeau and M. Schatzman, A wave problem in a half-space with a unilateral contraint at the boundary. J. Diff. Eq. 53 (1984) 309-361. | MR | Zbl

[23] A. Petrov and M. Schatzman, Viscoélastodynamique monodimensionnelle avec conditions de Signorini. C. R. Acad. Sci. Paris Sér. I 334 (2002) 983-988. | MR | Zbl

[24] L.Q. Qi and J. Sun, A nonsmooth version of Newton's method. Math. Program. 58 (1993) 353-367. | MR | Zbl

[25] M. Raous, L. Cangémi and M. Cocu, A consistent model coupling adhesion, friction, and unilateral contact. Comput. Methods Appl. Mech. Engrg. 177 (1999) 383-399. | MR | Zbl

[26] M. Schatzman, A hyperbolic problem of second order with unilateral constraints: the vibrating string with a concave obstacle. J. Math. Anal. Appl. 73 (1980) 138-191. | MR | Zbl

[27] M. Shillor, M. Sofonea and J. Telega, Models and Analysis of Quasistatic Contact, Lect. Notes Phys. 655. Springer, Berlin-Heidelberg-New York (2004). | Zbl

[28] 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). | MR | Zbl

[29] D.E. Stewart, Convolution complementarity problems with application to impact problems. IMA J. Appl. Math. 71 (2006) 92-119. | MR | Zbl

[30] D.E. Stewart, Differentiating complementarity problems and fractional index convolution complementarity problems. Houston J. Math. 33 (2007) 301-322. | MR | Zbl

[31] D.E. Stewart, Energy balance for viscoelastic bodies in frictionless contact. (Submitted).

[32] M.E. Taylor, Partial Differential Equations 1, Applied Mathematical Sciences 115. Springer-Verlag, New York (1996). | MR

[33] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam, New York (1978). | MR | Zbl

[34] J. Wloka, Partial Differential Equations. Cambridge University Press (1987). | MR | Zbl