A class of history-dependent differential variational inequalities with application to contact problems

ABSTRACT

In this paper a class of generalized differential variational inequalities with constraints involving history-dependent operators in Banach spaces is investigated. The unique solvability and regularity results are obtained via surjectivity of multivalued pseudomonotone operators combined with a fixed point principle. From abstract results, a theorem concerning existence, uniqueness and regularity of weak solution to a frictional viscoelastic contact problem with adhesion and history-dependent operator is established. Further, a theoretical analysis of a penalty method for history-dependent differential variational inequality is provided. The unique solvability of a penalized problem is shown, as well as the convergence of its solution to the solution of the original history-dependent differential variational inequality, as a penalty parameter tends to zero. Finally, results on a penalty method are applied to another contact problem, history-dependent frictional viscoelastic contact problem with a generalized normal compliance condition instead of a generalized Signorini contact condition.

KEYWORDS:

• History-dependent differential variational inequality
• Coulomb law of dry friction
• Signorini condition
• adhesion
• penalty method

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35L15
• 35L86
• 35L87
• 74Hxx
• 74M10

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH. It is supported by the National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262 and the National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. The first author is also supported by the NSF of Guangxi Grant No. 2018JJA110006, and Beibu Gulf University Project No. 2018KYQD06. The second author is also supported by NNSF of China Grant No. 11671101, and NSF of Guangxi Grant No. 2018GXNSFDA138002.