A thermoviscoelastic contact problem with friction, damage and wear diffusion

Afrika Matematika - Tập 34 - Trang 1-17 - 2023
Soumia Latreche1, Lynda Selmani2
1Department of Sciences, Teacher Education College of Setif, El-Eulma, Algeria
2Laboratory of Applied Mathematics, Faculty of Sciences, University Ferhat Abbas of Setif 1, Setif, Algeria

Tóm tắt

In this paper we present a model for quasistatic frictional contact between a thermoviscoelastic body and a moving foundation that involves wear of contacting surface and diffusion of wear debris. The damage effect is taken into account in the thermoviscoelastic constitutive law, its evolution is described by a parabolic inclusion with the homogeneous Neumann boundary condition. Contact is modeled with a normal compliance condition and is associated to a dry friction. The wear takes place on a part of the contact surface, when the wear debris surface density diffuse on the whole of the contact surface and is accompanied by frictional heat exchange. We derive a variational formulation of the problem and state that, under a smallness assumption on the problem data, there exists a unique weak solution for the model. The proof is based on elliptic variational inequalities, parabolic variational inequalities, first order evolution equations and fixed point arguments.

Tài liệu tham khảo

Andrews, K.T., Klarbring, A., Shillor, M., Wright, S.: A dynamic thermoviscoelastic contact problem with friction and wear. Int. J. Eng. Sci. 35(14), 1291–1309 (1997) Barbu, V.: Optimal Control of Variational Inequalities. Pitman, Boston (1984) Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1988) Figueiro, I., Trabucho, L.: A class of contact and friction dynamic problems in thermoelasticity and in thermoviscoelasticity. Int. J. Eng. Sci. 33(1), 45–66 (1995) Frémond, M.: Non-Smooth Thermomechanics. Springer, Berlin (2002) Frémond, M., Nedjar, B.: Damage in concrete: the unilateral phenomenon. Nuclear Eng. Des. 156(1–2), 323–335 (1995) Frémond, M., Nedjar, B.: Damage, gradient of damage and principle of virtual work. Int. J. Solids Struct. 33(8), 1083–1103 (1996) Gasiński, L., Ochal, A.: Dynamic thermoviscoelastic problem with friction and damage. Nonlinear Anal. Real World Appl. 21, 63–75 (2015) Gasiński, L., Ochal, A., Shillor, M.: Quasistatic thermoviscoelastic problem with normal compliance, multivalued friction and wear diffusion. Nonlinear Anal. Real World Appl. 27, 183–202 (2016) Gu, R.J., Shillor, M.: Thermal and wear analysis of an elastic beam in sliding contact. Int. J. Solids Struct. 38(14), 2323–2333 (2001) Gu, R.J., Kuttler, K.L., Shillor, M.: Frictional wear of a thermoelastic beam. J. Math. Anal. Appl. 242(2), 212–236 (2000) Johansson, L., Klarbring, A.: Thermoelastic frictional contact problems: modeling, finite element approximation and numerical realization. Comp. Methods Appl. Mech. Eng. 105(2), 181–210 (1993) Kuttler, K.L., Renard, Y., Shillor, M.: Models and simulations of dynamic frictional contact. Comp. Method Appl. Mech. Eng. 177(3–4), 259–272 (1999) Rochdi, M., Shillor, M.: Existence and uniqueness for a quasistatic frictional bilateral contact problem in thermoviscoelasticity. Q. Appl. Math. 58(3), 543–560 (2000) Rochdi, M., Shillor, M., Sofonea, M.: A quasistatic viscoelastic contact problem with normal compliance and friction. J. Elast. 51(2), 105–126 (1998) Selmani, M.: Frictional contact problem with wear for electro-viscoelastic materials with long memory. Bull. Belg. Math. Simon Stevin. 20(3), 461–479 (2013) Selmani, M., Selmani, L.: A frictional contact problem with wear and damage for electro-viscoelastic materials. Appl. Math. 55(2), 89–109 (2010) Shillor, M., Sofonea, M.: A quasistatic viscoelastic contact problem with friction. Int. J. Eng. Sci. 38(14), 1517–1533 (2000) Shillor, M., Sofonea, M., Telega, J.J.: Quasistatic viscoelastic contact with friction and wear diffusion. Quart. Appl. Math. 62(2), 379–399 (2004) Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact: Variational Methods. Lect. Notes Phys., vol. 655. Springer, Berlin Heidelberg (2004) Sofonea, M., Shillor, M.: Variational analysis of quasistatic viscoplastic contact problems with friction. Commun. Appl. Anal. 5(1), 135–151 (2001) Sofonea, M., Han, W., Shillor, M.: Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman-Hall/CRC Press, New York (2006) Strömberg, N., Johansson, L., Klarbring, A.: Derivation and analysis of a generalized standard model for contact friction and wear. Int. J. Solids Struct. 33(13), 1817–1836 (1996) Zeidler, E.: Nonlinear Functional Analysis and Its Applications, II/A: Linear Monotone Operators. Springer-Verlag, New York (1990)