A class of fractional  differential history-dependent hemivariational inequalities with application to thermo-viscoleastic

Zakaria Faiz; Hicham  Benaissa; Othmane Baiz; Driss El Moutawkil

doi:10.5269/bspm.68107

Zakaria Faiz faiz
Hicham Benaissa
Othmane Baiz
Driss El Moutawkil

Abstract

The aim of this work is to study a class of fractional differential history-dependent hemivariational inequalities and to provide an example of application in thermo-viscoelasticity. Using the Rothe method and the subjectivity property of multivalued pseudomonotone operators, we first prove existence of a solution. The proof is based on a fixed point argument and a recent finding from hemivariational inequalities theory. Then, we apply the obtained abstract results to a nonlinear thermo-viscoelastic contact problem with a historydependent with fractional time Kelvin-Voiget constitution law and adhesion.

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References

A. Amassad, C. Fabre, M. Sofonea, A quasistatic viscoplastic contact problem with normal compliance and friction. IMA J. Appl Math. 69, 463-482 (2004).

Z. Shengda, Z. Liu, S. Migorski, A class of fractional differential hemivariational inequalities with application to contact problem. Z. Angew. Math. Phys. 69, 36 (2018).

H. Xuan, X. Cheng, Numerical analysis of a thermal frictional contact problem with long memory. Communication on pure and applied analysis 20(4), 1521-1543 (2021).

J. Jarusek, M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems. Z. Angew Math Mech (ZAMM), 88(1), 3-22 (2008).

Z. Shengda, Z. Liu, S. Migorski, A class of time-fractional hemivariational Inequalities with application to frictional contact problem. Communications in Nonlinear Science and Numerical Simulation 56, 34-48 (2018).

S. Migorski, A. Ochal, M. Sofonea, History-dependent variational-hemivariational inequalities in contact mechanics. Nonlinear Anal. Real World Appl. 22, 6041-7618 (2015).

X. Li, Z. Liu, M. Sofonea, Unique solvability and exponential stability of differential hemi-variational inequalities. Applicable Analysis 99(14) (2020).

H. Benaissa, El-H. Essoufi, R. Fakhar, Existence results for unilateral contact problem with friction of thermoelectroelasticity. Appl. Math. Mech. 36, 9111-7926 (2015).

Z. Liu, S. Zeng, D. Motreanu, Partial differential hemivariational inequalities. Adv. Nonlinear Anal. 7(4), 571-586 (2018)

S. Migorski, A. Ochal, Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion. Nonlinear Anal. Theory Methods Appl. 69, 495-509 (2008).

S. Migorski, Z. Zeng, Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics. Numerical Algorithms 82, 423-450 (2019).

S. Migorski, A. Ochal, M. Sofonea, Nonlinear inclusions and hemivariational inequalities: Models and analysis of contact problems. , Advances in Mechanics and Mathematics, vol. 26, Springer, New York (2013).

M. Sofonea, S. Migorski, W. Han, A penalty method for history-dependent variational-hemivariational inequalities. Comput. Math. Appl. 75, 2561-2573 (2018).

J. Kacur, Method of Rothe in Evolution Equations. Teubner-Texte zur Mathematik, vol. 80. B.G. Teubner, Leipzig (1985).

M. Sofonea, W. Han, S. Migorski, Numerical analysis of history-dependent variational-hemivariational inequalities with applications to contact problems. Euro. J. Appl. Math. 26, 4271-7452 (2015).

R. Herrman, Fractional calculs: An Introduction for Physicists. World Scientific, Singapore (2011).

I. Podlubny, Fractional Differential Equations. Academic, San Diego (1999).

O. Chau, Numerical analysis of a thermal contact problem with adhesion. Comp. Appl. Math. 37, 5424-5455 (2018).

Z. Naniewicz, P.D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications. Marcel Dekker, Inc., New York (1995).

P.D. Panagiotopoulos, Nonconvex problems of semipermeable media and related topics. ZAMM Z. Angew. Math. Mech. 65: 29-36 (1985).

P.D. Panagiotopoulos, Hemivariational Inequalities, applications in mechanics and engineering. Springer, Berlin (1993).

W. Han, S. Migorski, M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 389-3912 (2014).