An elastic-viscoplastic contact problem with internal state variable, normal damped response and unilateral constraint
DOI:
https://doi.org/10.24193/subbmath.2025.2.11Keywords:
Contact problem, elastic-viscoplastic material, internal state variable, normal damped response, unilateral constraint, friction, quasivariational inequalityAbstract
In this manuscript, we study a contact problem between an elastic-viscoplastic body and an obstacle. The contact is quasistatic and it is described with a normal damped response condition with friction and unilateral constraint. Moreover, we use an elastic-viscoplastic constitutive law with internal state variable to model the material's behavior. We present the classical problem then we derive its variational formulation. Finally, we prove that the associated variational problem has a unique solution. The proof is based on arguments of quasivariational inequalities and fixed points.
References
[1] Barboteu, M., Danan, D., Sofonea, M., Analysis of a contact problem with normal damped response and unilateral constraint, ZAMM-Z Angew Math. Mech., 96(2015), no. 4, 408-428.
[2] Chau, O., Awbi, B., Quasistatic thermoviscoelastic frictional contact with damped response, Appl. Anal., 86(2004), no. 6, 635-648.
[3] Chouchane, L., Selmani, L., On a viscoplastic model coupling adhesion and damage, An. Univ. Oradea Fasc. Mat., 15(2008), 63-89.
[4] Chouchane, L., Selmani, L., A frictionless elastic-viscoplastic contact problem with normal comliance, adhesion and damage, Stud. Univ. Babe s-Bolyai Math., 54(2009), no. 1, 43-64.
[5] Chouchane, L., Selmani, L., A history-dependent frictional contact problem with wear for thermoviscoelastic materials, Math. Model. Anal., 24(2019), no. 3, 351-371.
[6] Eck, C., Jaru sek, J., Krbe c, M., Unilateral contact problems: Variational methods and existence theorems, Pure Applied Mathematics, Chapman/CRC Press, New York, 2005.
[7] Fern andez, J.R., Han, W., Via~no, J.M., Variational and numerical analysis of a frictionless contact problem for elastic-viscoplastic materials with internal state variables, Quart. J. Mech. Appl. Math., 54(2001), no. 4, 501-522.
[8] Latreche, S., Selmani, L., Analysis of a frictionless contact problem with adhesion for piezoelectric materials, Taiwanese J. Math., 21(2017), 81-105.
[9] Selmani, M., Selmani, L., A frictional contact problem with normal damped response for elastic-visco-plastic materials, Rend. Circ. Mat. Palermo, 59(2010), 67-85.
[10] Selmani, M., Selmani, L., Analysis of frictionless contact problem for elastic-viscoplastic materials, Nonlinear Anal. Model. Control, 17(2012), no. 1, 99-117.
[11] Sofonea, M., Bollati, J., Tarzia, D.A., Optimal control of di erential quasivariational inequalities with applications in contact mechanics, J. Math. Anal. Appl., 493(2021), no. 2, 124567.
[12] Sofonea, M., Chouchane, L., Quasistatic adhesive contact of piezoelectric cylinders, Nonlinear Anal. Model. Control, 14(2009), no. 1, 123-142.
[13] Sofonea, M., Chouchane, L., Selmani, L., Analysis of an antiplane contact problem with adhesion for electro-viscoelastic materials, Nonlinear Anal. Model. Control, 13(2008), no. 3, 379-395.
[14] M Sofonea, M., Han, W., Barboteu, M., Analysis of a viscoelastic contact problem with multivalued normal compliance and unilateral constraint, Comput. Methods Appl. Mech. Engrg., 264(2013), 12-22.
[15] Sofonea, M., Matei, A., Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, vol. 398, Cambridge University Press, Cambridge, 2012.
[16] Sofonea, M., P atrulescu, F., Farcas, A., A viscoplastic contact problem with normal compliance, unilateral constraint and memory term, Appl. Math. Optim., 69(2014), no. 2, 175-198.
[17] Sofonea, M., Souleimane, Y., A viscoelastic sliding contact problem with normal com- pliance, unilateral constraint and memory term, Mediterr. J. Math., 13(2016), no. 5, 2863-2886.
