Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints
This work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-12-01
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| Series: | Technologies |
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| Online Access: | https://www.mdpi.com/2227-7080/9/1/1 |
| _version_ | 1828417033432203264 |
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| author | Mircea Sofonea Meir Shillor |
| author_facet | Mircea Sofonea Meir Shillor |
| author_sort | Mircea Sofonea |
| collection | DOAJ |
| description | This work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem with unilateral constraints for elastic materials, which is governed by three parameters. A weak formulation of the problem is derived, which is in the form of an elliptic variational inequality, and the Tykhonov well-posedness of the problem is established, under appropriate assumptions on the data and parameters, with respect to a special Tykhonov triple. The proof is based on arguments on coercivity, compactness, and lower-semicontinuity. This abstract result leads to different convergence results, which establish the continuous dependence of the weak solution on the data and the parameters. Moreover, these results elucidate the links among the weak solutions of the different models. Finally, the corresponding mechanical interpretations of the conditions and the results are provided. The novelty in this work is the application of the Tykhonov well-posedness concept, which allows a unified and elegant framework for this class of static contact problems. |
| first_indexed | 2024-12-10T14:13:18Z |
| format | Article |
| id | doaj.art-728a5518f2b143bcb321cae0506a6a85 |
| institution | Directory Open Access Journal |
| issn | 2227-7080 |
| language | English |
| last_indexed | 2024-12-10T14:13:18Z |
| publishDate | 2020-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Technologies |
| spelling | doaj.art-728a5518f2b143bcb321cae0506a6a852022-12-22T01:45:26ZengMDPI AGTechnologies2227-70802020-12-0191110.3390/technologies9010001Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral ConstraintsMircea Sofonea0Meir Shillor1Laboratoire de Mathématiques et Physique, University of Perpignan, Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, FranceDepartment of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USAThis work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem with unilateral constraints for elastic materials, which is governed by three parameters. A weak formulation of the problem is derived, which is in the form of an elliptic variational inequality, and the Tykhonov well-posedness of the problem is established, under appropriate assumptions on the data and parameters, with respect to a special Tykhonov triple. The proof is based on arguments on coercivity, compactness, and lower-semicontinuity. This abstract result leads to different convergence results, which establish the continuous dependence of the weak solution on the data and the parameters. Moreover, these results elucidate the links among the weak solutions of the different models. Finally, the corresponding mechanical interpretations of the conditions and the results are provided. The novelty in this work is the application of the Tykhonov well-posedness concept, which allows a unified and elegant framework for this class of static contact problems.https://www.mdpi.com/2227-7080/9/1/1contact problemunilateral constraintvariational inequalityTykhonov tripleTykhonov well-posednessapproximating sequence |
| spellingShingle | Mircea Sofonea Meir Shillor Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints Technologies contact problem unilateral constraint variational inequality Tykhonov triple Tykhonov well-posedness approximating sequence |
| title | Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints |
| title_full | Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints |
| title_fullStr | Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints |
| title_full_unstemmed | Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints |
| title_short | Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints |
| title_sort | tykhonov well posedness and convergence results for contact problems with unilateral constraints |
| topic | contact problem unilateral constraint variational inequality Tykhonov triple Tykhonov well-posedness approximating sequence |
| url | https://www.mdpi.com/2227-7080/9/1/1 |
| work_keys_str_mv | AT mirceasofonea tykhonovwellposednessandconvergenceresultsforcontactproblemswithunilateralconstraints AT meirshillor tykhonovwellposednessandconvergenceresultsforcontactproblemswithunilateralconstraints |