Skip to main content
SpringerLink
Log in

Free Energy, Free Entropy, and a Gradient Structure for Thermoplasticity

  • Chapter
  • First Online:
Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 81))

Abstract

In the modeling of solids the free energy, the energy, and the entropy play a central role. We show that the free entropy, which is defined as the negative of the free energy divided by the temperature, is similarly important. The derivatives of the free energy are suitable thermodynamical driving forces for reversible (i.e. Hamiltonian) parts of the dynamics, while for the dissipative parts the derivatives of the free entropy are the correct driving forces. This difference does not matter for isothermal cases nor for local materials, but it is relevant in the non-isothermal case if the densities also depend on gradients, as is the case in gradient thermoplasticity. Using the total entropy as a driving functional, we develop gradient structures for quasistatic thermoplasticity, which again features the role of the free entropy. The big advantage of the gradient structure is the possibility of deriving time-incremental minimization procedures, where the entropy-production potential minus the total entropy is minimized with respect to the internal variables and the temperature. We also highlight that the usage of an auxiliary temperature as an integrating factor in [30] serves exactly the purpose to transform the reversible driving forces, obtained from the free energy, into the needed irreversible driving forces, which should have been derived from the free entropy. This reconfirms the fact that only the usage of the free entropy as driving functional for dissipative processes allows us to derive a proper variational formulation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alber, H. -D. (1998). Materials with memory (Vol. 1682), Lecture Notes in Mathematics. Berlin: Springer.

    Google Scholar 

  2. Ambrosio, L., Gigli, N., & Savaré, G. (2005). Gradient flows in metric spaces and in the space of probability measures., Lectures in Mathematics Basel: ETH Zürich. Birkhäuser Verlag.

    MATH  Google Scholar 

  3. Bartels, S., & Roubíček, T. (2008). Thermoviscoplasticity at small strains. ZAMM—Journal of Applied Mathematics and Mechanics, 88, 735–754.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bartels, S., & Roubíček, T. (2011). Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion. Mathematical Modelling and Numerical Analysis (M2AN), 45, 477–504.

    Google Scholar 

  5. Carathéodory, C. (1909). Untersuchungen über die Grundlagen der Thermodynamik. Mathematische Annalen, 67, 355–386.

    Article  MathSciNet  Google Scholar 

  6. Carstensen, C., Hackl, K., & Mielke, A. (2002). Non-convex potentials and microstructures in finite-strain plasticity. Proceedings of the Royal Society of London Series A, 458(2018), 299–317.

    Article  MathSciNet  MATH  Google Scholar 

  7. Dal Maso, G., DeSimone, A., & Mora, M. G. (2006). Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Archive for Rational Mechanics and Analysis, 180(2), 237–291.

    Google Scholar 

  8. Dal Maso, G., DeSimone, A., & Solombrino, F. (2011). Quasistatic evolution for cam-clay plasticity: a weak formulation via viscoplastic regularization and time parametrization. Calculus of Variations and Partial Differential Equations, 40(2), 125–181.

    Google Scholar 

  9. Gürses, E., Mainik, A., Miehe, C., & Mielke, A. (2006). Analytical and numerical methods for finite-strain elastoplasticity. In R. Helmig, A. Mielke, & B. Wohlmuth (Eds.), Multifield problems in solid and fluid mechanics (pp. 443–481). Berlin: Springer.

    Google Scholar 

  10. Gröger, K. (1978). Zur Theorie des quasi-statischen Verhaltens von elastisch-plastischen Körpern. ZAMM—Journal of Applied Mathematics and Mechanics, 58(2), 81–88.

    Article  MATH  Google Scholar 

  11. Hütter, M., & Svendsen, B. (2012). Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible-irreversible coupling. Continuum Mechanics and Thermodynamics, 24, 211–227.

    Article  MathSciNet  Google Scholar 

  12. Johnson, C. (1976). Existence theorems for plasticity problems. Journal de Mathematiques Pures et Appliques (9), 55(4), 431–444.

    Google Scholar 

  13. Mainik, A., & Mielke, A. (2009). Global existence for rate-independent gradient plasticity at finite strain. Journal of Nonlinear Science, 19(3), 221–248.

    Article  MathSciNet  MATH  Google Scholar 

  14. Miehe, C., & Stein, E. (1992). A canonical model of multiplicative elasto–plasticity. Formulation and aspects of numerical implementation. European Journal of Mechanics endash; A/Solids, 11, 25–43.

    Google Scholar 

  15. Mielke, A. (2003). Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Continuum Mechanics and Thermodynamics, 15, 351–382.

    Article  MathSciNet  MATH  Google Scholar 

  16. Mielke, A. (2011). Formulation of thermoelastic dissipative material behavior using GENERIC. Continuum Mechanics and Thermodynamics, 23(3), 233–256.

    Article  MathSciNet  MATH  Google Scholar 

  17. Mielke, A. (2011). On thermodynamically consistent models and gradient structures for thermoplasticity. GAMM—Mitteilungen, 34(1), 51–58.

    MathSciNet  MATH  Google Scholar 

  18. Mielke, A. (2016). On evolutionary \(\varGamma \)-convergence for gradient systems. In A. Muntean, J. Rademacher, & A. Zagaris (Eds.), Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity, Lecture Notes in Applied Math. Mechanics, 3, 187–249. Springer.

    Google Scholar 

  19. Mielke, A. (2013). Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete and Continuous Dynamical Systems—Series S, 6(2), 479–499.

    Article  MathSciNet  MATH  Google Scholar 

  20. Mielke, A., & Roubíček, T. (2015). Rate-independent systems: theory and application. Applied Mathematical Sciences, 193. Springer.

    Google Scholar 

  21. Mielke, A., & Stefanelli, U. (2015). Homogenizing the penrose-fife system via its gradient structure. In preparation.

    Google Scholar 

  22. Moreau, J.-J. (1974). On unilateral constraints, friction and plasticity. In New Variational Techniques in Mathematical Physics (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Bressanone, 1973) (pp. 171–322). Rome: Edizioni Cremonese.

    Google Scholar 

  23. Onsager, L. (1931). Reciprocal relations in irreversible processes, I+II. Physical Review, 37, 405–426. (part II, 38:2265–2279).

    Google Scholar 

  24. Ortiz, M., & Repetto, E. (1999). Nonconvex energy minimization and dislocation structures in ductile single crystals. Journal of the Mechanics and Physics of Solids, 47(2), 397–462.

    Article  MathSciNet  MATH  Google Scholar 

  25. Ortiz, M., & Stainier, L. (1999). The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering, 171(3–4), 419–444.

    Article  MathSciNet  MATH  Google Scholar 

  26. Ortiz, M., Repetto, E., & Stainier, L. (2000). A theory of subgrain dislocation structures. Journal of the Mechanics and Physics of Solids, 48, 2077–2114.

    Article  MathSciNet  MATH  Google Scholar 

  27. Penrose, O., & Fife, P. C. (1990). Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D, 43(1), 44–62.

    Article  MathSciNet  MATH  Google Scholar 

  28. Penrose, O., & Fife, P. C. (1993). On the relation between the standard phase-field model and a "thermodynamically consistent" phase-field model. Physica D, 69(1–2), 107–113.

    Article  MathSciNet  MATH  Google Scholar 

  29. Simo, J., & Ortiz, M. (1985). A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive relations. Computer Methods in Applied Mechanics and Engineering, 49, 221–245.

    Article  MATH  Google Scholar 

  30. Yang, Q., Stainier, L., & Ortiz, M. (2006). A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids. Journal of the Mechanics and Physics of Solids, 54, 401–424.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to Michael Ortiz for many stimulating discussions on the interaction of mathematics and mechanics. Moreover, he thanks Ulisse Stefanelli for the fruitful collaboration concerning variational methods for evolutionary systems. The resarch was partially supported by the European Research Council via the grant ERC-2010-AdG no.267802 Analysis of Multiscale Systems Driven by Functionals.

Author information

Authors and Affiliations

  1. WIAS, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

Authors
  1. Alexander Mielke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Mielke .

Editor information

Editors and Affiliations

  1. FB 11 - Maschinenbau, Universität Siegen, Siegen, Germany

    Kerstin Weinberg

  2. Politecnico di Milano, Milano, Italy

    Anna Pandolfi

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Mielke, A. (2016). Free Energy, Free Entropy, and a Gradient Structure for Thermoplasticity. In: Weinberg, K., Pandolfi, A. (eds) Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems. Lecture Notes in Applied and Computational Mechanics, vol 81. Springer, Cham. https://doi.org/10.1007/978-3-319-39022-2_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-39022-2_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-39021-5

  • Online ISBN: 978-3-319-39022-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation