June  2015, 35(6): 2349-2403. doi: 10.3934/dcds.2015.35.2349

Analysis of a model coupling volume and surface processes in thermoviscoelasticity

1. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia

2. 

DICATAM - Sezione di Matematica, Università di Brescia, Via Valotti 9, I-25133 Brescia, Italy, Italy

Received  February 2014 Revised  May 2014 Published  December 2014

We focus on a highly nonlinear evolutionary PDE system describing volume processes coupled with surfaces processes in thermoviscoelasticity, featuring the quasi-static momentum balance, the equation for the unidirectional evolution of an internal variable on the surface, and the equations for the temperature in the bulk domain and the temperature on the surface. A significant example of our system occurs in the modeling for the unidirectional evolution of adhesion between a body and a rigid support, subject to thermal fluctuations and in contact with friction.
    We investigate the related initial-boundary value problem, and in particular the issue of existence of global-in-tim solutions, on an abstract level. This allows us to highlight the analytical features of the problem and, at the same time, to exploit the tight coupling between the various equations in order to deduce suitable estimates on (an approximation of) the problem.
    Our existence result is proved by passing to the limit in a carefully tailored approximate problem, and by extending the obtained local-in-time solution by means of a refined prolongation argument.
Citation: Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Analysis of a model coupling volume and surface processes in thermoviscoelasticity. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2349-2403. doi: 10.3934/dcds.2015.35.2349
References:
[1]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984.

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

[3]

G. Bonfanti, F. Luterotti and M. Frémond, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24.

[4]

E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion, Math. Meth. Appl. Sci., 31 (2008), 1029-1064. doi: 10.1002/mma.957.

[5]

E. Bonetti, G. Bonfanti and R. Rossi, Well-posedness and long-time behaviour for a model of contact with adhesion, Indiana Univ. Math. J., 56 (2007), 2787-2819. doi: 10.1512/iumj.2007.56.3079.

[6]

E. Bonetti, G. Bonfanti and R. Rossi, Thermal effects in adhesive contact: Modelling and analysis, Nonlinearity, 22 (2009), 2697-2731. doi: 10.1088/0951-7715/22/11/007.

[7]

E. Bonetti, G. Bonfanti and R. Rossi, Long-time behaviour of a thermomechanical model for adhesive contact, Discrete Contin. Dyn. Syst. Ser. S., 4 (2011), 273-309. doi: 10.3934/dcdss.2011.4.273.

[8]

E. Bonetti, G. Bonfanti and R. Rossi, Analysis of a unilateral contact problem taking into account adhesion and friction, J. Differential Equations, 253 (2012), 438-462. doi: 10.1016/j.jde.2012.03.017.

[9]

E. Bonetti, G. Bonfanti and R. Rossi, Analysis of a temperature-dependent model for adhesive contact with friction, Phys. D, 285 (2014), 42-62. doi: 10.1016/j.physd.2014.06.008.

[10]

E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Global solution to a singular integrodifferential system related to the entropy balance, Nonlinear Anal., 66 (2007), 1949-1979. doi: 10.1016/j.na.2006.02.035.

[11]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Number 5 in North Holland Math. Studies. North-Holland, Amsterdam, 1973.

[12]

H. Brézis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., 1 (2001), 387-404. doi: 10.1007/PL00001378.

[13]

E. Di Benedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751. doi: 10.1137/0512062.

[14]

G. Duvaut, Équilibre d'un solide élastique avec contact unilatéral et frottement de Coulomb, C. R. Acad. Sci. Paris Sér. A-B., 290 (1980), A263-A265.

[15]

C. Eck, J. Jarušek and M. Krbec, Unilateral Contact Problems, Pure and Applied Mathematics (Boca Raton) Vol. 270, Chapman & Hall/CRC, Boca Raton, Florida, 2005. doi: 10.1201/9781420027365.

[16]

M. Frémond and B. Nedjar, Damage, gradient of damage and priciple of virtual power, Internat. J. Solids Structures, 33 (1996), 1083-1103. doi: 10.1016/0020-7683(95)00074-7.

[17]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.

[18]

M. Frémond, Phase Change in Mechanics, Springer-Verlag, Berlin Heidelberg, 2012.

[19]

A. D. Ioffe, On lower semicontinuity of integral functionals, I. SIAM J. Control Optimization (4), 15 (1977), 521-538. doi: 10.1137/0315035.

[20]

J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[21]

M. Sofonea, W. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.

[22]

M. Valadier, Young measures, in Methods of nonconvex analysis (Varenna, 1989), Lecture Notes in Math., Springer, Berlin, 1446 (1990), 152-188. doi: 10.1007/BFb0084935.

[23]

A. Visintin, Models of Phase Transitions, Progress in Nonlinear Differential Equations and their Applications, 28, Birkhauser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4078-5.

show all references

References:
[1]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984.

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

[3]

G. Bonfanti, F. Luterotti and M. Frémond, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24.

[4]

E. Bonetti, G. Bonfanti and R. Rossi, Global existence for a contact problem with adhesion, Math. Meth. Appl. Sci., 31 (2008), 1029-1064. doi: 10.1002/mma.957.

[5]

E. Bonetti, G. Bonfanti and R. Rossi, Well-posedness and long-time behaviour for a model of contact with adhesion, Indiana Univ. Math. J., 56 (2007), 2787-2819. doi: 10.1512/iumj.2007.56.3079.

[6]

E. Bonetti, G. Bonfanti and R. Rossi, Thermal effects in adhesive contact: Modelling and analysis, Nonlinearity, 22 (2009), 2697-2731. doi: 10.1088/0951-7715/22/11/007.

[7]

E. Bonetti, G. Bonfanti and R. Rossi, Long-time behaviour of a thermomechanical model for adhesive contact, Discrete Contin. Dyn. Syst. Ser. S., 4 (2011), 273-309. doi: 10.3934/dcdss.2011.4.273.

[8]

E. Bonetti, G. Bonfanti and R. Rossi, Analysis of a unilateral contact problem taking into account adhesion and friction, J. Differential Equations, 253 (2012), 438-462. doi: 10.1016/j.jde.2012.03.017.

[9]

E. Bonetti, G. Bonfanti and R. Rossi, Analysis of a temperature-dependent model for adhesive contact with friction, Phys. D, 285 (2014), 42-62. doi: 10.1016/j.physd.2014.06.008.

[10]

E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Global solution to a singular integrodifferential system related to the entropy balance, Nonlinear Anal., 66 (2007), 1949-1979. doi: 10.1016/j.na.2006.02.035.

[11]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Number 5 in North Holland Math. Studies. North-Holland, Amsterdam, 1973.

[12]

H. Brézis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., 1 (2001), 387-404. doi: 10.1007/PL00001378.

[13]

E. Di Benedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751. doi: 10.1137/0512062.

[14]

G. Duvaut, Équilibre d'un solide élastique avec contact unilatéral et frottement de Coulomb, C. R. Acad. Sci. Paris Sér. A-B., 290 (1980), A263-A265.

[15]

C. Eck, J. Jarušek and M. Krbec, Unilateral Contact Problems, Pure and Applied Mathematics (Boca Raton) Vol. 270, Chapman & Hall/CRC, Boca Raton, Florida, 2005. doi: 10.1201/9781420027365.

[16]

M. Frémond and B. Nedjar, Damage, gradient of damage and priciple of virtual power, Internat. J. Solids Structures, 33 (1996), 1083-1103. doi: 10.1016/0020-7683(95)00074-7.

[17]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.

[18]

M. Frémond, Phase Change in Mechanics, Springer-Verlag, Berlin Heidelberg, 2012.

[19]

A. D. Ioffe, On lower semicontinuity of integral functionals, I. SIAM J. Control Optimization (4), 15 (1977), 521-538. doi: 10.1137/0315035.

[20]

J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[21]

M. Sofonea, W. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics (Boca Raton), 276, Chapman & Hall/CRC, Boca Raton, FL, 2006.

[22]

M. Valadier, Young measures, in Methods of nonconvex analysis (Varenna, 1989), Lecture Notes in Math., Springer, Berlin, 1446 (1990), 152-188. doi: 10.1007/BFb0084935.

[23]

A. Visintin, Models of Phase Transitions, Progress in Nonlinear Differential Equations and their Applications, 28, Birkhauser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4078-5.

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