A contact problem in thermoviscoelastic diffusion theory with second sound

Aouadi, Moncef; Copetti, Maria I.M.; Fernández, José R.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 759-796.

We consider a contact problem in thermoviscoelastic diffusion theory in one space dimension with second sound. The contact is modeled by the Signorini’s condition and the stress-strain constitutive equation is of Kelvin−Voigt type. The thermal and diffusion disturbances are modeled by Cattaneo’s law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier’s law. The system of equations is a coupling of a hyperbolic equation with four parabolic equations. It poses some new mathematical difficulties due to the nonlinear boundary conditions and the lack of regularity. We prove that the viscoelastic term provides additional regularity leading to the existence of weak solutions. Then, fully discrete approximations to a penalized problem are considered by using the finite element method. A stability property is shown, which leads to a discrete version of the energy decay property. A priori error analysis is then provided, from which the linear convergence of the algorithm is derived. Finally, we give some computational results.

Classification : 65N30, 65N15
Mots clés : Thermoviscoelastic, diffusion, contact, existence, exponential stability, numerical analysis

M. Aouadi, Theory of generalized micropolar thermoelastic diffusion under Lord-Shulman model. J. Thermal Stresses 32 (2009) 923–942. | DOI

M. Aouadi, The coupled theory of micropolar thermoelastic diffusion. Acta Mech. 208 (2009) 181–203. | DOI | Zbl

M. Aouadi, A theory of thermoelastic diffusion materials with voids. Z. Angew. Math. Phys. 61 (2010) 357–379. | DOI | MR | Zbl

M. Aouadi, A contact problem of a thermoelastic diffusion rod. Z. Angew. Math. Mech. 90 (2010) 278–286. | DOI | MR | Zbl

M. Aouadi, Classic and Generalized Thermoelastic Diffusion Theories. Encyclopedia of Thermal Stresses. Springer Science+Business, Media Dordrecht (2014).

M. Aouadi, On thermoelastic diffusion thin plates theory. Appl. Math. Mech. -Engl. Ed. 36 (2015) 619–632. | DOI | MR | Zbl

M. Aouadi and M.I.M. Copetti, Analytical and numerical results for a dynamic contact problem with two stops in thermoelastic diffusion theory. Z. Angew. Math. Mech. 96 (2016) 361–384. | DOI | MR

M. Aouadi, B. Lazzari and R. Nibbi, A theory of thermoelasticity with diffusion under Green-Naghdi models. Z. Angew. Math. Mech. 94 (2014) 837–852. | DOI | MR | Zbl

M. Barboteu, J.R. Fernández and T.V. Hoarau-Mantel, A class of evolutionary variational inequalities with applications in viscoelasticity. Math. Models Methods Appl. Sci. 15 (2005) 1595–1617. | DOI | MR | Zbl

A. Berti, M.I.M. Copetti, J.R. Fernández and M.G. Naso, A dynamic thermoviscoelastic contact problem with the second sound effect. J. Math. Anal. Appl. 421 (2015) 1163–1195. | DOI | MR | Zbl

M. Campo, J.R. Fernández, K.L. Kuttler and M. Shillor, Quasistatic evolution of damage in an elastic body: numerical analysis and computational experiments. Appl. Numer. Math. 57 (2007) 975–988. | DOI | MR | Zbl

P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Vol. 4 of Studies in Mathematics and its Applications. North-Holland Publishing, Amsterdam (1978). | MR | Zbl

M.I.M. Copetti, A contact problem in generalized thermoelasticity. Appl. Math. Comput. 218 (2011) 2128–2145. | MR | Zbl

M.I.M. Copetti and M. Aouadi, Analysis of a contact problem in thermoviscoelasticity under the Green−Lindsay model. Appl. Num. Math. 91 (2015) 60–74. | DOI | MR | Zbl

C.M. Elliott and Q. Tang, A dynamic contact problem in thermoelasticity. Nonlinear Anal. 23 (1994) 883–898. | DOI | MR | Zbl

H. Fernandez Sare and R. Racke, On the stability of damped Timoshenko systems–Cattaneo versus Fourier law. Arch. Ration. Mech. Anal. 194 (2009) 221–251. | DOI | MR | Zbl

J.U. Kim, A boundary thin obstacle problem for a wave equation. Comm. Partial Differ. Eq. 14 (1989) 1011–1026. | DOI | MR | Zbl

J. Kubik, Thermodiffusion in viscoelastic solids. Studia Geot. Mech. 8 (1986) 29–47.

J. Kubik, A thermodynamics derivation of constitutive relations of thermodiffusion in Kelvin−Voigt medium. Acta Mech. 146 (2001) 135–138. | DOI | Zbl

K.L. Kuttler and M. Shillor, Vibrations of a beam between two stops. Dyn. Contin. Discrete Impulsive Syst. Ser. B 8 (2001) 93–110. | MR | Zbl

H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15 (1967) 299–309. | DOI | Zbl

J. Munoz Rivera and R. Racke, Multidimensional contact problems in thermoelasticity. SIAM J. Appl. Math. 58 (1998) 1307–1337. | DOI | MR | Zbl

E. Rabinowicz, Friction and Wear of Materials, 2nd edition. Wiley, New York (1995).

H.H. Sherief, F. Hamza and H. Saleh, The theory of generalized thermoelastic diffusion. Int. J. Eng. Sci. 42 (2004) 591–608. | DOI | MR | Zbl

P. Shi and M. Shillor, Existence of a solution to the $n$-dimensional problem of thermoelastic contact. Comm. Partial Differ. Eq. 17 (1992) 1597–1618. | DOI | MR | Zbl

J. Sprenger, On a contact problem in thermoelasticity with second sound. Quart. Appl. Math. 67 (2009) 601–615. | DOI | MR | Zbl

L. Thomas, Fundamentals of Heat Transfer. Prentice-Hall Inc., Englewood Cliffs, New Jersey (1980).