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A Mortar Method Combined with an Augmented Lagrangian Approach for Treatment of Mechanical Contact Problems

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Multibody Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 35))

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Abstract

This work presents a mixed penalty-duality formulation from an augmented Lagrangian approach for treating the contact inequality constraints. The augmented Lagrangian approach allows to regularize the non differentiable contact terms and gives a C\(^1\) differentiable saddle-point functional. The relative displacement of two contacting bodies is described in the framework of the Finite Element Method (FEM) using the mortar method, which gives a smooth representation of the contact forces across the bodies interface. To study the robustness and performance of the proposed algorithm, validation numerical examples with finite deformations and large slip motion are presented.

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Acknowledgments

This work has received financial support from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT), Universidad Nacional del Litoral (CAI+D 2009 PI65-330).

Author information

Authors and Affiliations

  1. Centro de Investigación de Métodos Computacionales (CIMEC), Universidad Nacional del Litoral—CONICET, Predio Conicet-Santa Fe, Colectora Ruta Nac 168 s/n/ Paraje El Pozo, 3000, Santa Fe, Argentina

    Federico J. Cavalieri & Alberto Cardona

  2. Department of Aerospace and Mechanical Engineering (LTAS), University of Liège, Chemin des Chevreuils, 1 (B52), 4000, Liège, Belgium

    Olivier Brüls

Authors
  1. Federico J. Cavalieri
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  2. Olivier Brüls
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  3. Alberto Cardona
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Corresponding author

Correspondence to Alberto Cardona .

Editor information

Editors and Affiliations

  1. Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia

    Zdravko Terze

Appendix

Appendix

The linearization of the tangential vector \({{\varvec{t}}}_A\) is presented. The tangential vector \({{\varvec{t}}}_A\) is used in the slip status, thus

$$\begin{aligned} {{\varvec{t}}}_A= \frac{\varvec{\sigma }_{TA}}{\parallel \varvec{\sigma }_{TA}\parallel } = \frac{\varvec{\sigma }_{TA}}{-\mu \sigma _{NA}}. \end{aligned}$$
(4.43)

The linearization operator \(\varDelta \) applied to Eq. (4.43), yields

$$\begin{aligned} \varDelta {{\varvec{t}}}_A = \frac{[{{\varvec{I}}}-{{\varvec{t}}}_A\otimes {{\varvec{t}}}_A]\varDelta \varvec{\sigma }_{A}}{\parallel \varvec{\sigma }_{TA}\parallel } = \frac{[{{\varvec{I}}}-{{\varvec{t}}}_A\otimes {{\varvec{t}}}_A]\varDelta \varvec{\sigma }_{A}}{-\mu \sigma _{NA}}. \end{aligned}$$
(4.44)

After some algebraic manipulations the linearization of the tangential vector is written as

$$\begin{aligned} \varDelta {{\varvec{t}}}_A = -\frac{{{\varvec{I}}}-{{\varvec{t}}}_A\otimes {{\varvec{t}}}_A - \varvec{\nu }_A\otimes \varvec{\nu }_A}{\mu \sigma _{NA}}\varDelta \varvec{\sigma }_A + \frac{\varvec{\nu }_A\otimes \varvec{\sigma }_A + ({{\varvec{I}}}- {{\varvec{t}}}_A\otimes {{\varvec{t}}}_A) \sigma _{NA} }{\mu \sigma _{NA}}\varDelta \varvec{\nu }_A. \end{aligned}$$
(4.45)

If the variation of the normal vector \(\varvec{\nu }_A\) is neglected, the final expression is given by

$$\begin{aligned} \varDelta {{\varvec{t}}}_A = -\frac{{{\varvec{I}}}-{{\varvec{t}}}_A\otimes {{\varvec{t}}}_A - \varvec{\nu }_A\otimes \varvec{\nu }_A}{\mu \sigma _{NA}}\varDelta \varvec{\sigma }_A. \end{aligned}$$
(4.46)

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Cavalieri, F.J., Brüls, O., Cardona, A. (2014). A Mortar Method Combined with an Augmented Lagrangian Approach for Treatment of Mechanical Contact Problems. In: Terze, Z. (eds) Multibody Dynamics. Computational Methods in Applied Sciences, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-319-07260-9_4

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  • DOI: https://doi.org/10.1007/978-3-319-07260-9_4

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  • Online ISBN: 978-3-319-07260-9

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