Volume 2, Issue 3
A Perturbation Result for Dynamical Contact Problems

Corinna Klapproth, Peter Deuflhard & Anton Schiela

Numer. Math. Theor. Meth. Appl., 2 (2009), pp. 237-257.

Published online: 2009-02

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  • Abstract

This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.

  • Keywords

Dynamical contact problems, stability, (visco-)elasticity, Signorini condition, Newmark method.

  • AMS Subject Headings

35L85, 74H55, 74M15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-2-237, author = {}, title = {A Perturbation Result for Dynamical Contact Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2009}, volume = {2}, number = {3}, pages = {237--257}, abstract = {

This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2009.m9003}, url = {http://global-sci.org/intro/article_detail/nmtma/6024.html} }
TY - JOUR T1 - A Perturbation Result for Dynamical Contact Problems JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 237 EP - 257 PY - 2009 DA - 2009/02 SN - 2 DO - http://doi.org/10.4208/nmtma.2009.m9003 UR - https://global-sci.org/intro/article_detail/nmtma/6024.html KW - Dynamical contact problems, stability, (visco-)elasticity, Signorini condition, Newmark method. AB -

This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.

Corinna Klapproth, Peter Deuflhard & Anton Schiela. (2020). A Perturbation Result for Dynamical Contact Problems. Numerical Mathematics: Theory, Methods and Applications. 2 (3). 237-257. doi:10.4208/nmtma.2009.m9003
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