An efficient approach for numerical treatment of some inequalities in solid mechanics on examples of Kuhn–Tucker and Signorini–Fichera conditions

A computationally efficient solution scheme is presented for the mechanical problems whose formulations include the Kuhn–Tucker or Signorini–Fichera conditions. It is proposed to reformulate these problems replacing inequalities in these conditions by equations with respect to new unknowns. The solu...

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Bibliographic Details
Published in:Journal of the mechanics and physics of solids 2014-02, Vol.63, p.432-450
Main Authors: Kindrachuk, Vitaliy M., Galanov, Boris A.
Format: Article
Language:English
Subjects:
Algorithms
Computational efficiency
Computing time
Constitutive behavior
Contact mechanics
Crystal plasticity
Inclusions
Inequalities
Kuhn–Tucker and Signorini–Fichera conditions
Mathematical analysis
Mathematical models
Nonlinearity
Numerical algorithms
Operators
Solid mechanics
Solvers
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Summary:A computationally efficient solution scheme is presented for the mechanical problems whose formulations include the Kuhn–Tucker or Signorini–Fichera conditions. It is proposed to reformulate these problems replacing inequalities in these conditions by equations with respect to new unknowns. The solutions of the modified problems have simple physical meanings and determine uniquely the unknowns of the original problems. The approach avoids application of multi-valued operators (inclusions or inequalities) in formulation of the problems. Hence, the modified formulations are suitable for numerical analysis using established powerful mathematical methods and corresponding solvers developed for solving systems of non-linear equations. To demonstrate the advantages of the proposed approach, it is applied for solving problems in two different areas: constitutive modeling of single-crystal plasticity and mixed boundary value problems of elastic contact mechanics with free boundaries. The original formulations of these problems contain respectively the Kuhn–Tucker and Signorini–Fichera conditions. A problem of the former area is integrated using an implicit integration scheme based on the return-mapping algorithm. The derived integration scheme is free of any update procedure for identification of active slip systems. A problem of the latter area is reduced to solution of non-linear integral boundary equations (NBIEs). Numerical examples demonstrate stability and efficiency of the solution procedures and reflect the mathematical similarities between the both non-linear problems.
ISSN:0022-5096
DOI:10.1016/j.jmps.2013.08.008