ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
Proceeding cover
proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Thermodynamic consistency of third grade finite strain elasticity; pp. 126–132
PDF | doi: 10.3176/proc.2010.2.10

Authors
Péter Ván, Christina Papenfuss
Abstract
The thermodynamic framework of finite strain viscoelasticity with second order weak nonlocality in the deformation gradient is investigated. The application of Liu’s procedure leads to a class of third grade elastic materials where the second gradient of the stress appears in the elastic constitutive relation. Finally the dispersion relation of longitudinal plane waves is calculated in isotropic materials.
References

  1. Truesdell, C. and Noll, W. The Non-Linear Field Theories of Mechanics. Springer, Berlin, 1965.

  2. Gurtin, M. E. Thermodynamics and the possibility of spatial interaction in elastic materials. Arch. Ration. Mech. Anal., 1965, 19, 339–352.
doi:10.1007/BF00253483

  3. Acharya, A. and Shawki, T. G. Thermodynamic restrictions on constitutive equations for second-deformation-gradient inelastic behaviour. J. Mech. Phys. Solids, 1995, 43, 1751–1772.
doi:10.1016/0022-5096(95)00054-M

  4. Papenfuss, C. and Forest, S. Thermodynamical frameworks for higher grade material theories with internal variables or additional degrees of freedom. J. Non-Equil. Thermodyn., 2006, 31, 319–353.

  5. Gurtin, M. E. Configurational Forces as Basic Concepts of Continuum Physics. Springer, New York, 2000.

  6. Maugin, G. The Thermomechanics of Nonlinear Irreversible Behaviors (An introduction). World Scientific, Singapore, 1999.
doi:10.1142/9789812796271

  7. Ván, P. Weakly nonlocal irreversible thermodynamics. Ann. Phys.-Leipzig, 2003, 12, 146–173.

  8. Liu, I-Shih. Entropy flux relation for viscoelastic bodies. J. Elasticity, 2008, 90, 259–270.
doi:10.1007/s10659-007-9142-0

  9. Jou, D., Casas-Vázquez, J., and Lebon, G. Extended Irreversible Thermodynamics. Springer, Berlin, 2001.

10. Müller, I. and Ruggeri, T. Rational Extended Thermodynamics. Springer, New York, 1998.

11. Verhás, J. Thermodynamics and Rheology. Akadémiai Kiadó and Kluwer Academic Publishers, Budapest and Dordrecht, 1997.

12. Triani, V., Papenfuss, C., Cimmelli, V. A., and Muschik, W. Exploitation of the Second Law: Coleman-Noll and Liu procedure in comparison. J. Non-Equil. Thermodyn., 2008, 33, 47–60.

13. Muschik, W., Triani, V., and Papenfuss, C. Exploitation of the dissipation inequality, if some balances are missing. J. Mech. Mat. Struct., 2008, 3, 1125–1133.
doi:10.2140/jomms.2008.3.1125

14. Ván, P. Weakly nonlocal non-equilibrium thermodynamics – variational principles and Second Law. In Applied Wave Mathematics (Quak, E. and Soomere, T., eds). Springer, Berlin, 2009, 153–186.
doi:10.1007/978-3-642-00585-5_10

15. Aifantis, E. C. Update on a class of gradient theories. Mech. Mat., 2003, 35, 259–280.
doi:10.1016/S0167-6636(02)00278-8

16. Noll, W. A mathematical theory of the mechanical behavior of continuous media. Arch. Rat. Mech. Anal., 1958/59, 2, 197–226.
doi:10.1007/BF00277929

17. Matolcsi, T. and Ván, P. Can material time derivative be objective? Phys. Lett. A, 2006, 353, 109–112.
doi:10.1016/j.physleta.2005.12.072

18. Matolcsi, T. and Ván, P. Absolute time derivatives. J. Mat. Phys., 2007, 48, 053507–19.
doi:10.1063/1.2719144

19. Fülöp, T. A new interpretation of the kinematics of continua. In New Results in Continuum Physics (Fülöp, T., ed.), Vol. 8 of Notes on Engineering Geology and Rock Mechanics. BME Publisher, Budapest, 2008, Chapter 3, 55–99 (in Hungarian).

20. Ván, P. Material manifolds in nonrelativistic spacetime. In New Results in Continuum Physics (Fülöp, T., ed.), Vol. 8. Mérnökgeológia-Kőzetmechanika Kiskönyvtár. Publishing House of the Budapest University of Technology and Economics, Budapest, 2008, 37–54 (in Hungarian).

21. Pawłow, I. Thermodynamically consistent Cahn–Hilliard and Allen–Cahn models in elastic solids. Disc. Cont. Dyn. Syst., 2006, 15, 1169–1191.
doi:10.3934/dcds.2006.15.1169

22. Ván, P. Internal energy in dissipative relativistic fluids. J. Mech. Mat. Struct., 2008, 3, 1161–1169.
doi:10.2140/jomms.2008.3.1161

23. Ván, P. Exploiting the Second Law in weakly nonlocal continuum physics. Period. Polytech. Mech., 2005, 49, 79–94.

24. Cimmelli, V. A. An extension of Liu procedure in weakly nonlocal thermodynamics. J. Math. Phys., 2007, 48, 113510.
doi:10.1063/1.2804753

25. Mindlin, R. D. Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct., 1965, 1, 417–438.
doi:10.1016/0020-7683(65)90006-5

26. Porubov, A. V., Aero, E. L., and Maugin, G. A. Two approaches to study essential nonlinear and dispersive properties of the internal structure of materials. Phys. Rev. E, 2009, 79, 046608.
doi:10.1103/PhysRevE.79.046608

27. Berezovski, A., Engelbrecht, J., and Maugin, G. A. Generalized thermomechanics with dual internal variables. Arch. Appl. Mech., 2010,
doi: 10.1007/s00419-010-0412-0.

28. Forest, S. and Sievert, R. Nonlinear microstrain theories. Int. J. Solids Struct., 2006, 43, 7224–7245.
doi:10.1016/j.ijsolstr.2006.05.012
Back to Issue