Advertisement

Use and Abuse of the Method of Virtual Power in Generalized Continuum Mechanics and Thermodynamics

  • Samuel Forest
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)

Abstract

The method of virtual power, put forward by Paul Germain and celebrated by Gérard A. Maugin, is used (and abused) in the present work in combination with continuum thermodynamics concepts in order to develop generalized continuum, phase field, higher order temperature and diffusion theories. The systematic and effective character of the method is illustrated in the case of gradient and micromorphic plasticity models. It is then tentatively applied to the introduction of temperature and concentration gradient effects in diffusion theories leading to generalized heat and mass diffusion equations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aifantis E (1984) On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology 106:326–330Google Scholar
  2. Altenbach H, Maugin GA, Erofeev V (2011) Mechanics of Generalized Continua, Advanced Structured Materials, vol 7. Springer, HeidelbergGoogle Scholar
  3. Ammar K, Appolaire B, Cailletaud G, Feyel F, Forest S (2009) Finite element formulation of a phase field model based on the concept of generalized stresses. Computational Materials Science 45:800–805Google Scholar
  4. Aslan O, Forest S (2011) The micromorphic versus phase field approach to gradient plasticity and damage with application to cracking in metal single crystals. In: de Borst R, Ramm E (eds) Multiscale Methods in Computational Mechanics, Lecture Notes in Applied and Computational Mechanics 55, Springer, pp 135–154Google Scholar
  5. Aslan O, Cordero NM, Gaubert A, Forest S (2011) Micromorphic approach to single crystal plasticity and damage. International Journal of Engineering Science 49:1311–1325Google Scholar
  6. Cahn J, Hilliard J (1958) Free energy of a nonuniform system. I. Interfacial free energy. The Journal of Chemical Physics 28:258–267Google Scholar
  7. Coleman B, Mizel J (1963) Thermodynamics and departures from Fourier’s law of heat conduction. Arch Rational Mech and Anal 13:245–261Google Scholar
  8. Coleman B, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Rational Mech and Anal 13:167–178Google Scholar
  9. Daher N, Maugin GA (1986) The method of virtual power in continuum mechanics application to media presenting singular surfaces and interfaces. Acta Mechanica 60:217–240Google Scholar
  10. dell’Isola F, Seppecher P (1995) The relationship between edge contact forces, double forces and intersticial working allowed by the principle of virtual power. CR Acad Sci Paris IIb 321:303–308Google Scholar
  11. dell’Isola F, Seppecher P, Madeo A (2012) How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach “à la D’Alembert”. Zeitschrift für Angewandte Mathematik und Physik 63:1119–1141Google Scholar
  12. Finel A, Le Bouar Y, Gaubert A, Salman U (2010) Phase field methods: Microstructures, mechanical properties and complexity. Comptes Rendus Physique 11:245–256Google Scholar
  13. Forest S (2008) The micromorphic approach to plasticity and diffusion. In: Jeulin D, Forest S (eds) Continuum Models and Discrete Systems 11, Proceedings of the international conference CMDS11, Les Presses de l’Ecole des Mines de Paris, Paris, France, pp 105–112Google Scholar
  14. Forest S (2009) The micromorphic approach for gradient elasticity, viscoplasticity and damage. ASCE Journal of Engineering Mechanics 135:117–131Google Scholar
  15. Forest S (2016) Nonlinear regularisation operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 472(2188)Google Scholar
  16. Forest S, Aifantis EC (2010) Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. International Journal of Solids and Structures 47:3367–3376Google Scholar
  17. Forest S, Amestoy M (2008) Hypertemperature in thermoelastic solids. Comptes Rendus Mécanique 336:347–353Google Scholar
  18. Forest S, Bertram A (2011) Formulations of strain gradient plasticity. In: Altenbach H, Maugin GA, Erofeev V (eds) Mechanics of Generalized Continua, Advanced Structured Materials vol. 7, Springer, pp 137–150Google Scholar
  19. Forest S, Sab K (2012) Continuum stress gradient theory. Mechanics Research Communications 40:16–25Google Scholar
  20. Forest S, Sab K (2017) Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models. Mathematics and Mechanics of SolidsGoogle Scholar
  21. Frémond M, Nedjar B (1996) Damage, gradient of damage and principle of virtual power. Int J Solids Structures 33:1083–1103Google Scholar
  22. Germain P (1973a) La méthode des puissances virtuelles en mécanique des milieux continus, première partie : théorie du second gradient. J de Mécanique 12:235–274Google Scholar
  23. Germain P (1973b) The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J Appl Math 25:556–575Google Scholar
  24. Germain P, Nguyen QS, Suquet P (1983) Continuum thermodynamics. Journal of Applied Mechanics 50:1010–1020Google Scholar
  25. Gurtin M (1996) Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D 92:178–192Google Scholar
  26. Gurtin M (2003) On a framework for small–deformation viscoplasticity: free energy, microforces, strain gradients. International Journal of Plasticity 19:47–90Google Scholar
  27. Gurtin M, Anand L (2009) Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck & Hutchinson and their generalization. Journal of the Mechanics and Physics of Solids 57:405–421Google Scholar
  28. Ireman P, Nguyen QS (2004) Using the gradients of temperature and internal parameters in continuum mechanics. CR Mécanique 332:249–255Google Scholar
  29. LiuW, Saanouni K, Forest S, Hu P (2017) The micromorphic approach to generalized heat equations. Journal of Non-Equilibrium Thermodynamics 42(4):327–358Google Scholar
  30. Maugin GA (1980) The method of virtual power in continuum mechanics: Application to coupled fields. Acta Mechanica 35:1–70Google Scholar
  31. Maugin GA (1990) Internal variables and dissipative structures. J Non–Equilib Thermodyn 15:173–192Google Scholar
  32. Maugin GA (1992) Thermomechanics of Plasticity and Fracture. Cambridge University PressGoogle Scholar
  33. Maugin GA (1999) Thermomechanics of Nonlinear Irreversible Behaviors. World ScientificGoogle Scholar
  34. Maugin GA (2006) On the thermomechanics of continuous media with diffusion and/or weak nonlocality. Archives of Applied Mechanics 75:723–738Google Scholar
  35. Maugin GA (2013) The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool. Continuum Mechanics and Thermodynamics 25:127–146Google Scholar
  36. Maugin GA, Metrikine AV (eds) (2010) Mechanics of Generalized Continua - One Hundred Years After the Cosserats, Advances in Mechanics and Mathematics, vol 21. Springer, New YorkGoogle Scholar
  37. Maugin GA, Muschik W (1994) Thermodynamics with internal variables, Part I. General concepts. J Non-Equilib Thermodyn 19:217–249Google Scholar
  38. Mindlin R (1965) Second gradient of strain and surface–tension in linear elasticity. Int J Solids Structures 1:417–438Google Scholar
  39. Nguyen QS (2010a) Gradient thermodynamics and heat equations. Comptes Rendus Mécanique 338:321–326Google Scholar
  40. Nguyen QS (2010b) On standard dissipative gradient models. Annals of Solid and Structural Mechanics 1(2):79–86Google Scholar
  41. Nguyen QS (2016) Quasi-static responses and variational principles in gradient plasticity. Journal of the Mechanics and Physics of Solids 97:156–167Google Scholar
  42. Saanouni K, Hamed M (2013) Micromorphic approach for finite gradient-elastoplasticity fully coupled with ductile damage: Formulation and computational aspects. International Journal of Solids and Structures 50:2289–2309Google Scholar
  43. Svendsen B (1999) On the thermodynamics of thermoelastic materials with additional scalar degrees of freedom. Continuum Mechanics and Thermodynamics 4:247–262Google Scholar
  44. Temizer I, Wriggers P (2010) A micromechanically motivated higher–order continuum formulation of linear thermal conduction. ZAMM 90:768–782Google Scholar
  45. Villani A, Busso E, Ammar K, Forest S, Geers M (2014) A fully coupled diffusional-mechanical formulation: numerical implementation, analytical validation, and effects of plasticity on equilibrium. Archive of Applied Mechanics 84:1647–1664Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Centre des Matériaux, CNRS, UMR 7633Mines ParisTechParisFrance

Personalised recommendations