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Numerical solution of frictional contact problems based on a mortar algorithm with an augmented Lagrangian technique

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Abstract

This work presents a frictional contact formulation to solve three-dimensional contact problems with large finite displacements. The kinematic description of the contacting bodies is defined by using a mortar approach. The regularization of the variational frictional contact problem is solved with a mixed dual penalty approach based on an augmented Lagrangian technique. In this method, the numerical results do not depend on the definition of any user-defined penalty parameter affecting the normal or tangential component of forces. The robustness and performance of the proposed algorithm are studied and validated by solving a series of numerical examples with finite displacements and large slip.

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Acknowledgements

This work has received financial support from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), PIP 2011/01105, and from Universidad Nacional del Litoral (CAI+D 2011).

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

    F. J. Cavalieri & A. Cardona

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Appendix

Appendix

The linearization of the tangential vector t A is presented. The tangential vector t is used in the slip status, thus

$$ \boldsymbol {t}_A= \frac{\boldsymbol {\sigma }_{\mathit{TA}}}{\|\boldsymbol {\sigma }_{\mathit{TA}}\| } = \frac{\boldsymbol {\sigma }_{\mathit{TA}}}{-\mu\sigma_{\mathit{NA}}}. $$
(38)

The linearization operator Δ applied to Eq. (38) yields

$$ \Delta \boldsymbol {t}_A = \frac{[\boldsymbol {I}-\boldsymbol {t}_A\otimes \boldsymbol {t}_A]\Delta \boldsymbol {\sigma }_{A}}{\|\boldsymbol {\sigma }_{\mathit{TA}}\|} = \frac{[\boldsymbol {I}-\boldsymbol {t}_A\otimes \boldsymbol {t}_A]\Delta \boldsymbol {\sigma }_{A}}{-\mu\sigma_{\mathit{NA}}}. $$
(39)

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

$$ \Delta \boldsymbol {t}_A = -\frac{\boldsymbol {I}-\boldsymbol {t}_A\otimes \boldsymbol {t}_A - \boldsymbol {\nu }_A\otimes \boldsymbol {\nu }_A}{\mu\sigma_{\mathit{NA}}}\Delta \boldsymbol {\sigma }_A + \frac {\boldsymbol {\nu }_A\otimes \boldsymbol {\sigma }_A + (\boldsymbol {I}- \boldsymbol {t}_A\otimes \boldsymbol {t}_A) \sigma_{\mathit{NA}} }{\mu\sigma_{\mathit{NA}}}\Delta \boldsymbol {\nu }_A. $$
(40)

If the variation of the normal vector ν A is neglected, i.e., if the normal vector is computed at the previous time step, the final expression is given by

$$ \Delta \boldsymbol {t}_A = -\frac{\boldsymbol {I}-\boldsymbol {t}_A\otimes \boldsymbol {t}_A - \boldsymbol {\nu }_A\otimes \boldsymbol {\nu }_A}{\mu\sigma_{\mathit{NA}}}\Delta \boldsymbol {\sigma }_A. $$
(41)

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Cavalieri, F.J., Cardona, A. Numerical solution of frictional contact problems based on a mortar algorithm with an augmented Lagrangian technique. Multibody Syst Dyn 35, 353–375 (2015). https://doi.org/10.1007/s11044-015-9449-8

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