Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-ssw5r Total loading time: 0.23 Render date: 2022-08-17T10:02:15.991Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints

Published online by Cambridge University Press:  07 March 2016

RICCARDO SCALA
Affiliation:
Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy email: riccardo.scala@unipv.it, giusch04@unipv.it Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany email: riccardo.scala@wias-berlin.de
GIULIO SCHIMPERNA
Affiliation:
Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy email: riccardo.scala@unipv.it, giusch04@unipv.it

Abstract

We consider a viscoelastic body occupying a smooth bounded domain $\Omega\subset \mathbb{R}^3$ under the effect of a volumic traction force g. The macroscopic displacement vector from the equilibrium configuration is denoted by u. Inertial effects are considered; hence, the equation for u contains the second-order term utt. On a part ΓD of the boundary of Ω, the body is fixed and no displacement may occur; on a second part ΓN ⊂ ∂Ω, the body can move freely; on a third portion ΓC ⊂ ∂Ω, the body is in adhesive contact with a solid support. The boundary forces acting on ΓC due to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a non-linear ordinary differential equation settled on ΓC and describing the evolution of the delamination order parameter z. Following the lines of a new approach outlined in Bonetti et al. (2015, arXiv:1503.01911) and based on duality methods in Sobolev–Bochner spaces, we define a suitable concept of weak solution to the resulting system of partial differential equations. Correspondingly, we prove an existence result on finite-time intervals of arbitrary length.

Type
Papers
Copyright
Copyright © Cambridge University Press 2016 

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

[1] Attouch, H. (1984) Variational Convergence for Functions and Operators, Pitman, London.Google Scholar
[2] Barboteu, M., Bartosz, K. & Kalita, P. (2015) A dynamic viscoelastic contact problem with normal compliance, finite penetration and nonmonotone slip rate dependent friction. Nonlinear Anal. Real World Appl. 22, 452472.CrossRefGoogle Scholar
[3] Barbu, V., Colli, P., Gilardi, G. & Grasselli, M. (2000) Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation. Differ. Integral Equ. 13, 12331262.Google Scholar
[4] Barbu, V. (1976) Nonlinear Semigroups and Differential Equations in Banach Spaces, Noord-hoff, Leyden.CrossRefGoogle Scholar
[5] Blanchard, D., Damlamian, A. & Ghidouche, H. (1989) A nonlinear system for phase change with dissipation. Differ. Integral Equ. 2, 344362.Google Scholar
[6] Bonetti, E., Bonfanti, G. & Rossi, R. (2008) Global existence for a contact problem with adhesion. Math. Methods Appl. Sci. 31, 10291064.CrossRefGoogle Scholar
[7] Bonetti, E., BonfantiG. & Rossi, R. (2012) Analysis of a unilateral contact problem taking into account adhesion and friction. J. Differ. Equ. 235, 438462.CrossRefGoogle Scholar
[8] Bonetti, E., Rocca, E., Scala, R. & Schimperna, G. (2015) On the strongly damped wave equation with constraint, arXiv:1503.01911, submitted.Google Scholar
[9] Brézis, H. (1973) Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Studies, Vol. 5, North-Holland, Amsterdam.Google Scholar
[10] Brézis, H. (1972) Intégrales convexes dans les espaces de Sobolev. Israel J. Math. 13, 923.CrossRefGoogle Scholar
[11] Cocou, M., Schryve, M. & Raous, M. (2010) A dynamic unilateral contact problem with adhesion and friction in viscoelasticity. Z. Angew. Math. Phys. 61, 721743.CrossRefGoogle Scholar
[12] Cocou, M. (2015) A class of dynamic contact problems with Coulomb friction in viscoelasticity. Nonlinear Anal. Real World Appl. 22, 508519.CrossRefGoogle Scholar
[13] Colli, P., Luterotti, F., Schimperna, G. & Stefanelli, U. (2002) Global existence for a class of generalized systems for irreversible phase changes. NoDEA Nonlinear Differ. Equ. Appl. 9, 255276.CrossRefGoogle Scholar
[14] Frémond, M. (2012) Phase Change in Mechanics, Springer-Verlag, Berlin, Heidelberg.CrossRefGoogle Scholar
[15] Grun-Rehomme, M. (1977) Caractérisation du sous-différentiel d'intégrandes convexes dans les espaces de Sobolev (French). J. Math. Pures Appl. (9), 56, 149156.Google Scholar
[16] Ioffe, A. D. (1977) On lower semicontinuity of integral functionals. I. SIAM J. Control Optim. 15, 521538.CrossRefGoogle Scholar
[17] Kuttler, K. L., Menike, R. S. R. & Shillor, M. (2009) Existence results for dynamic adhesive contact of a rod. J. Math. Anal. Appl. 351, 781791.CrossRefGoogle Scholar
[18] Raous, M., Cangémi, L. & Cocu, M. (1999) A consistent model coupling adhesion, friction, and unilateral contact. Comput. Methods Appl. Mech. Eng. 177, 383399.CrossRefGoogle Scholar
[19] Rossi, R. & Roubíček, T. (2011) Thermodynamics and analysis of rate-independent adhesive contact at small strains. Nonlinear Anal. 74, 31593190.CrossRefGoogle Scholar
[20] Rossi, R. & Roubíček, T. (2013) Adhesive contact delaminating at mixed mode, its thermodynamics and analysis. Interfaces Free Bound. 15, 137.CrossRefGoogle Scholar
[21] Rossi, R. & Thomas, M. (2015) From an adhesive to a brittle delamination model in thermo-visco-elasticity. ESAIM Control Optim. Calc. Var. 21, 159.CrossRefGoogle Scholar
[22] Roubíček, T. (2005) Nonlinear Partial Differential Equations with Applications, Birkhäuser, Springer, Basel.Google Scholar
[23] Roubíček, T. (2013) Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity. SIAM J. Math. Anal. 45, 101126.CrossRefGoogle Scholar
[24] Scala, R. (2014) Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish, to appear on ESAIM:COCV.Google Scholar
[25] Schimperna, G. & Pawłow, I. (2013) On a class of Cahn-Hilliard models with nonlinear diffusion. SIAM J. Math. Anal. 45, 3163.CrossRefGoogle Scholar
[26] Simon, J. (1987) Compact sets in the space Lp (0, T;B). Ann. Mat. Pura Appl. 146, 6596.CrossRefGoogle Scholar
6
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *