Modeling and numerical analysis of a viscoelastic contact problem for locking materials with time-fractional derivatives
-
Jinxia Cen
und
Abdesadik Bendarag
Abstract
This article explores a new contact problem involving viscoelastic materials locked with a rigid foundation. The constitutive relation in our study is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. The contact is characterized by a variation of the Signorini condition, while friction is modeled by a nonlocal Coulomb’s friction law. We present the mathematical model for the viscoelastic process, provide the variational formulation, and establish the existence of a solution. We present fully discrete finite element schemes for the variational problem and derive error estimates for the approximations.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12371312
Funding statement: This project has received funding from the Natural Science Foundation of Guangxi Grant Nos. 2021GXNSFFA196004 and 2024GXNSFBA010337, the National Natural Science Foundation of China Grant No. 12371312, the Natural Science Foundation of Chongqing Grant No. CSTB2024NSCQ-JQX0033. It is also supported by the Postdoctoral Fellowship Program of CPSF under Grant Number GZC20241534, the Systematic Project of Center for Applied Mathematics of Guangxi (Yulin Normal University) Nos. 2023CAM002 and 2023CAM003.
A Appendix
Now, let us revisit the established definitions in fractional calculus theory and nonlinear analysis, given in references [14, 16, 27, 22].
Definition A.1 (Riemann–Liouville fractional integral).
Let X be a Banach space and
where
To further elaborate on the definition, we introduce
Definition A.2 (Caputo derivative of order,
0
<
α
≤
1
).
Let X be a Banach space,
The notation
It is obvious that if
Lemma A.3.
Let
for a positive constant c independent of N or k. Then there exists a positive constant c, independent of N and k, such that
Acknowledgements
The authors wish to thank the two knowledgeable referees for their corrections and helpful remarks.
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