Skip to main content
SpringerLink
Log in

A mechanical model for heat conduction

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

A theory of heat conduction in rigid heat conductors based entirely on mechanical concepts is proposed and compared with the traditional thermodynamic theories.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. This is the terminology used by Kestin and Rice [25]. As representatives of the two theories they cite, among others, the book [7] by de Groot and Mazur and the paper [4] by Coleman and Noll, respectively.

  2. Truesdell and Toupin [48], Sect. 245. For more comments of the same tenor, see Truesdell [45] and Marsden and Hughes [26], p. 176.

  3. Coleman and Noll [4].

  4. “The principle of material indifference is not a law of physics ...It is merely a prescription for avoiding nonsense,” Noll [35]. That indifference has nothing to do with physical reality but only with its representation is explained in detail in the paper [10].

  5. Curiously enough, though energy balance is a general law of physics, its position in classical continuum mechanics has never been clear. For example, in the theory of elasticity this law is ignored or, rather, is used to characterize the subclass of the hyperelastic materials. This was done by Truesdell and Noll [47], in spite of their mentioning energy balance among the “axioms of continuum physics” (ibid., Sect. 1), and claiming the necessity of its introduction as “an additional physical principle ...if definite problems are to be solved” (ibid., Sect. 45).

  6. For the recoverable part, the name free energy would be appropriate. Unfortunately, this term is generally used to denote the Helmholtz free energy, which is the recoverable part of the internal energy only in the special case of isothermal processes, see Remark 4.4 below. Therefore, denoting here by “free energy” the overall recoverable energy may generate confusion.

  7. The dissipative part of the internal energy is specified by the dissipation function, and the dissipation potential is the time derivative of that function.

  8. “... le comportement d’un milieu continu se trouve décrit essentiellement par deux fonctions à valeurs réelles ...un potentiel thermodynamique d’une part, une fonction des dissipations - ou plus généralement un quasi-potentiel des dissipations - d’autre part,” Germain [15], Introduction to Chapter VII. In his autobiographic paper [18], the same author says that what led him to this reformulation were Ziegler’s description of the different transpositions of Rayleigh’s dissipation function to viscoelasticity and to plasticity [49], Moreau’s enunciation of the lois de résistance and fonctions de résistance for friction, plasticity and viscosity [28], and Kestin’s “deep understanding of what may be an extension of thermostatics to thermodynamics” [25].

  9. Halphen and Nguyen [22].

  10. This deduction is by no means trivial. I was not able to find anywhere the assertion that the equation of virtual power is a consequence of energy conservation. On the contrary, that this equation has been proposed as the main axiom of continuum mechanics [16, 17] suggests that equation of virtual power and energy conservation are usually considered as independent concepts.

  11. See, e.g., Truesdell [46], p. 154.

  12. Mathematical definitions and axioms for a universe were enounced by Noll [32] and Truesdell ([46], Sect. I.2). Clearly they did not have in mind the physical universe but its representations, which they called material universes and universes of bodies, respectively.

  13. In the terminology of Germain ([15], Sect. VII.3), the integral in (5) is the dissipation function, and the function \(\chi \) is the pseudo-potential of dissipation. Here, for simplicity, I call \(\chi \) a dissipation potential.

  14. In fact, the assumption of differentiability of \(\chi \) is too restrictive, since it was clear from the beginning [28] that there are at least two types of dissipation potentials, one differentiable, and one with only directional derivatives. Examples of the two cases are viscosity and plasticity, respectively. In plasticity, \(\chi \) is not differentiable at the origin, and this leads to evolutions governed by an activation condition and by a flow rule, see, e.g., [9, 10]. In the viscous-like response assumed here the evolutions go on smoothly, without activation conditions and flow rules. Note that, while the potential \(\varphi \) is a function of state, the dissipation potential is not, because the integral in (5) is path-dependent.

  15. Note that \(\delta \vartheta \) is a perturbation of \({\dot{\vartheta }}_t\) and not of \(\vartheta _t\). This is essential to establish the form of the virtual power associated with both a “viscous” and a “plastic” dissipation potential. For the latter, see the paper [9].

  16. The cut principle excludes, for example, the presence of different surface effects at the physical boundary and at the interior surfaces. Note that while the contact actions s at the boundary of \(\Omega \) are accessible and, at least in principle, can be measured, the contact actions \(s_{\partial \Pi }\) at the interior points are inaccessible and therefore cannot be measured.

  17. \(\mathscr {V}\) is the vector space associated with \(\mathscr {E}\).

  18. This parallels the situation met in classical continuum mechanics, in which the mechanical contact action \(s_{\partial \Pi }(x)\) is a vector and the right-hand side of (11) becomes equal to \(T(x)\,n\), with T the stress tensor. In that context, the first equality in (11) was assumed by Cauchy and proved subsequently by Noll ([30], Theorem IV). The second equality is the thesis of Cauchy’s “tetrahedron theorem.” For more details, see [8], Sect. 4, and [10], Sect. 6.2.

  19. Due to the minus sign in (11), the flux is positive when q points inward \(\Pi \).

  20. This inequality states that the direction of the heat vector is opposite to the direction of the temperature gradient. It is preserved if, more in general, we take \(\chi \) independent of \(\nabla \dot{\vartheta }\) and \(\varphi (\vartheta ,\,\cdot \,)\) convex, nonnegative, and null at \(\nabla \vartheta =0\). Indeed, from the monotonicity of the gradient of a convex function, we have

    $$\begin{aligned} \big (\varphi _{\,\nabla \vartheta }(\vartheta ,\nabla \vartheta ) -\varphi _{\,\nabla \vartheta }(\vartheta ,\nabla \vartheta _0)\big )\cdot \big (\nabla \vartheta -\nabla \vartheta _0\big )\ge 0, \end{aligned}$$

    and in particular for \(\nabla \vartheta _0=0\), we get \(\varphi _{\,\nabla \vartheta }(\vartheta ,\nabla \vartheta )\cdot \nabla \vartheta \ge 0\). On the other hand, \(\varphi _{\,\nabla \vartheta }(\vartheta ,\nabla \vartheta )\) is equal to \(-q\) by (13)\(_2\) with \(\chi \) independent of \(\nabla \dot{\vartheta }\), whence inequality (18) follows.

  21. The terms involving \(k_0\) and \(c_0\) are necessary for reasons of physical plausibility. Indeed, for \(k_0=0\) the recoverable energy of a spatially uniform temperature field \(\nabla \vartheta =0\) would be zero at all temperatures. That is, the recoverable energy would be insensitive to uniform heating or cooling. Also, for \(c_0=0\) there would be no dissipation in a uniform change in temperature \(\nabla \dot{\vartheta }=0\).

  22. Straughan [44], Equation (1.37).

  23. The elimination of this term was not conveniently justified.

  24. See Straughan’s book [44].

  25. Green and Naghdi [20]. The variable \(\alpha \) was introduced by Boltzmann, who called it the thermal displacement. For this model, see also Giorgi et al. [19] and Bargmann et al. [1], and for the thermal displacement see Podio Guidugli [39, 41] and the papers cited therein.

  26. See Bargmann et al. [1], Sect. 6.

  27. Truesdell [45], Lecture 1.

  28. There are many formulations of the second law, including Kelvin’s and Clausius’ postulates, see [13] Sect. 7, and the Clausius–Duhem inequality, see below. The formulation (36) is given in Truesdell [45].

  29. The proof is based on the use of auxiliary Carnot cycles. See Planck [38], Sect. 128 or Fermi [13], Sects. 11–13 and 17.

  30. From comparison with (35) it is clear that, just like the internal energy, the entropy is determined to within an additive constant.

  31. See, e.g., [21] or [41]. According to [41], Eq. (1.7), the entropy imbalance consists in assuming the existence of a lower bound D for the entropy rate \(\dot{H}\). This implies the non-negativeness of the entropy production\(\varDelta =\dot{H}-D\). Comparing with Eq. (38) above, in which \(\varDelta \) is called \(\Gamma \), we see that D corresponds to \(Q/\varTheta \). By Eq. (1.9) and assumption (1.14) of [41], \(D=Q/\varTheta \) takes the form (41). The same Eq. (41) can be deduced from [21], Eqs. (27.7) to (27.14). For the assumption (1.14) of [41] about the form of the volume density of D, see the next Section 3.3.

  32. For the first equation, we recall that in a rigid conductor the mechanical working is zero.

  33. This inequality, called the condition of entropy imbalance in Gurtin et al. ([21], Sect. 27), is considered the formulation of the second law most appropriate to continua. See also Truesdell and Toupin [48] Sect. 258, and Coleman and Noll [4].

  34. See [4, 41], and point (ii) in Sect. 3.3 below.

  35. The Gibbs relation is usually considered a constitutive assumption, see, e.g., (Truesdell and Toupin [48], eq. (247.5)). The deduction of the relations (45) from the Clausius–Duhem inequality is considered an example of application of the Coleman–Noll procedure [4] for the formulation of constitutive equations and dissipation inequalities compatible with the two basic axioms.

  36. See [21], Sect. 27 or [41], Sect. 1.8.1.

  37. See [13], eq. (75).

  38. See ibid, equations (86), (87). Note that, by time differentiation, this equation yields \(\dot{\eta } ={\tilde{c}}\dot{\vartheta }/\vartheta \). In view of the Gibbs relation (45)\(_1\), this implies \(\dot{e}={\tilde{c}}\,\dot{\vartheta }\), in accordance with [41], eq. (1.41).

  39. Truesdell [45], p. 20.

  40. Gurtin et al. [21], Sect. 27.2.

  41. Coleman and Noll [4], Truesdell and Noll [47], Sect. 79.

  42. In [29], Müller assumes independent densities and shows that proportionality holds only for materials with a special symmetry group. In [40], Podio Guidugli restricts the general validity of the proportionality assumption to isotropic heat conductors. See also [41], Sect. 1.8.1.

  43. For cases in which this principle is violated, see [21], Sect. 66 and [47], Sect. 96.

  44. This does not exclude the dependence of \(\varphi \) and \(\chi \) on supplementary internal state variables, that is, on variables which do not appear in the external power. An example is classical plasticity, which involves an internal state variable, the plastic deformation. For more on this subject, see the paper [9].

  45. Gurtin et al. [21], Sect. 39.

  46. A “virtual power format for thermomechanics” was proposed by Podio Guidugli [39], see also [41] Chapter 4. His approach differs from the present one by the fact that in the expression (1) of the external power, the temperature rate \(\dot{\vartheta }_t\) is replaced by the time derivative \(\dot{\alpha }_t =\vartheta _t\) of the thermal displacement.

  47. Truesdell and Toupin [48], incipit of Chapter E.III.

  48. “In equilibrium thermodynamics ...  the state variables are usually independent of the space coordinates,” de Groot and Mazur [7], p. 3. See also Podio Guidugli [41], Sect. 1.7.1. In the language of Truesdell’s school, this notion of thermal equilibrium corresponds to the thermodynamics of homogeneous processes, see [48], Sect. 260, or [45], Sect. 1.

  49. Truesdell and Toupin ([48], Sect. 262). This is also Ziegler’s definition of reversibility ([50], Sect. 4.1).

  50. Gurtin et al. [21], incipit of Part V.

  51. Noll [31, 34].

  52. Noll [33].

  53. Noll [34].

  54. Truesdell and Toupin [48], Sect. 262.

References

  1. Bargmann, S., Favata, A., Podio Guidugli, P.: A revised exposition of the Green-Naghdi theory of heat propagation. J. Elast. 114, 143–154 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. Lond. A–458, 299–317 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Cattaneo, C.: Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3, 83–101 (1948)

    MathSciNet  MATH  Google Scholar 

  4. Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963)

    MathSciNet  MATH  Google Scholar 

  5. Dal Maso, G., De Simone, A., Mora, M.G.: Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180, 237–291 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Dal Maso, G., De Simone, A., Mora, M.G., Morini, M.: Globally stable quasistatic evolution problems in plasticity with softening. Netw. Heterog. Media 3, 567–614 (2008)

    MathSciNet  MATH  Google Scholar 

  7. de Groot, S.R., Mazur, P.: Non-equilibrium Themodynamics. Dover Publications, New York (1984). (Noth-Holland, Amsterdam 1962)

    Google Scholar 

  8. Del Piero, G.: Nonclassical continua, pseudobalance, and the law of action and reaction. Math. Mech. Complex Syst. 2, 71–107 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Del Piero, G.: The variational structure of classical plasticity. Math. Mech. Complex Syst. 6, 137–180 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Del Piero, G.: Reality and representation in mechanics: the legacy of Walter Noll. J. Elast. 135, 117–148 (2019)

    MathSciNet  MATH  Google Scholar 

  11. Edelen, D.G.B.: On the existence of symmetry relations and dissipation potentials. Arch. Ration. Mech. Anal. 51, 218–227 (1973)

    MathSciNet  MATH  Google Scholar 

  12. Fedelich, B., Ehrlacher, A.: Sur un principe de minimum concernant des matériaux à comportement indépendant du temps physique. C. R. Acad. Sci. Paris 308, 1391–1394 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Fermi, E.: Thermodynamics. Dover Publications, New York (1956). (Prentice-Hall 1937)

    MATH  Google Scholar 

  14. Francfort, G.A., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

    ADS  MathSciNet  MATH  Google Scholar 

  15. Germain, P.: Cours de Mécanique des Milieux Continus. Masson, Paris (1973)

    MATH  Google Scholar 

  16. Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus. Premiére partie: théorie du second gradient. J. Méc. 12, 235–274 (1973)

    MATH  Google Scholar 

  17. Germain, P.: The method of virtual power in continuum mechanics. Part 2: microstructure, SIAM J. Appl. Math. 25, 556–575 (1973)

    MATH  Google Scholar 

  18. Germain, P.: My discovery of mechanics. In: Maugin, G.A., Drouot, R., Sidoroff, F. (eds.) Continuum Thermomechanics, P. Germain’s Anniversary Volume, pp. 1–24. Kluwer, New York (2002)

    Google Scholar 

  19. Giorgi, C., Grandi, D., Pata, V.: On the Green-Naghdi type III heat conduction model. Discrete Contin. Dyn. Syst. B–19, 2133–2143 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A–432, 171–194 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  21. Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  22. Halphen, B., Nguyen, Q.S.: Sur les matériaux standards généralisés. J. Méc. 14, 39–63 (1975)

    MATH  Google Scholar 

  23. Hill, R., Rice, J.R.: Elastic potentials and the structure of inelastic constitutive laws. SIAM J. Appl. Math. 25, 448–461 (1973)

    MATH  Google Scholar 

  24. Houlsby, G.T., Puzrin, A.M.: A thermomechanical framework for constitutive models for rate-independent dissipative materials. Int. J. Plast. 16, 1017–1047 (2000)

    MATH  Google Scholar 

  25. Kestin, J., Rice, J.R.: Paradoxes in the application of thermodynamics to strained solids. In: Stuart, E.B. (ed.) A Critical Review of Thermodynamics. Mono Book Corp, Baltimore (1970)

    Google Scholar 

  26. Marsden, J.E., Hughes, J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983). (Reprinted by Dover Books, Mineola, N.Y. (1994))

  27. Mielke, A.: Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15, 351–382 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  28. Moreau, J.-J.: Sur les lois de frottement, de plasticité et de viscosité, C. R. Acad. Sc. Paris 271, 608–611 (1970)

    Google Scholar 

  29. Müller, I.: On the entropy inequality. Arch. Ration. Mech. Anal. 26, 118–141 (1967)

    MathSciNet  MATH  Google Scholar 

  30. Noll, W.: The foundations of classical mechanics in the light of recent advances in continuum mechanics. In: “The Axiomatic Method, with Special Reference to Geometry and Physics”, Symposium at Berkeley, 1957, pp. 266–281. North-Holland, Amsterdam (1959). (Reprinted in [37])

  31. Noll, W.: La mécanique classique, basée sur un axiome d’objectivité. In: “La Méthode Axiomatique dans les Mécaniques Classiques et Nouvelles”, Symposium in Paris, 1959, pp. 47-56, Gauthier-Villars, Paris (1963). (Reprinted in [37])

  32. Noll, W.: Lectures on the foundations of continuum mechanics and thermodynamics. Arch. Ration. Mech. Anal. 52, 62–92 (1973). (Reprinted in [37])

    MathSciNet  MATH  Google Scholar 

  33. Noll, W.: On material frame-indifference (1995). http://repository.cmu.edu/math/580

  34. Noll, W.: Updating “The Non-Linear Field Theories of Mechanics” (2004). http://www.math.cmu.edu/~wn0g/noll

  35. Noll, W.: On the past and future of natural philosophy. J. Elast. 84, 1–11 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Noll, W.: The Foundations of Continuum Mechanics and Thermodynamics. Selected Papers of W. Noll. Springer, Berlin (1974)

    MATH  Google Scholar 

  37. Petryk, H.: Incremental energy minimization in dissipative solids. C. R. Acad. Sci. Paris 331, 469–474 (2003)

    MATH  Google Scholar 

  38. Planck, M.: Treatise on Thermodynamics. Dover, Mineola (1945). (Translated from the 7th German edition 1922)

    Google Scholar 

  39. Podio Guidugli, P.: A virtual power format for thermomechanics. Contin. Mech. Thermodyn. 20, 479–487 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  40. Podio Guidugli, P.: On energy and entropy inflows in the theory of heat conduction. Arch. Appl. Mech. 85, 347–353 (2015)

    ADS  MATH  Google Scholar 

  41. Podio Guidugli, P.: Continuum Thermodynamics. SISSA Springer Series. Springer, Berlin (2019)

    MATH  Google Scholar 

  42. Rice, J.R.: On the structure of stress-strain relations for time-dependent plastic deformation in metals. J. Appl. Mech. 37, 728–737 (1970)

    ADS  Google Scholar 

  43. Rice, J.R.: Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 (1971)

    ADS  MATH  Google Scholar 

  44. Straughan, B.: Heat Waves. Springer, New York (2011)

    MATH  Google Scholar 

  45. Truesdell, C.: Rational Thermodynamics. McGraw-Hill, New York (1969)

    MATH  Google Scholar 

  46. Truesdell, C.: A First Course in Rational Continuum Mechanics, vol. 1, 2nd edn. Academic Press, Boston (1991)

    MATH  Google Scholar 

  47. Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/3. Springer, Berlin (1965)

    Google Scholar 

  48. Truesdell, C., Toupin, R.A.: The classical field theories. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/1. Springer, Berlin (1960)

    Google Scholar 

  49. Ziegler, H.: Some extremum principles in irreversible thermodynamics, with application to continuum mechanics. In: Sneddon, I.N., Hill, R. (eds.) Progress in Solid Mechanics, vol. 4, pp. 91–193. North Holland, Amsterdam (1963)

    Google Scholar 

  50. Ziegler, H.: An Introduction to Thermomechanics, 2nd edn. North Holland, Amsterdam (1983)

    MATH  Google Scholar 

Download references

Acknowledgements

I thank Paolo Podio Guidugli for stimulating discussions, and for his comments and suggestions.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria, Università di Ferrara, 44100, Ferrara, Italy

    Gianpietro Del Piero

  2. International Research Center M&MoCS, Cisterna di Latina, Italy

    Gianpietro Del Piero

Authors
  1. Gianpietro Del Piero
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gianpietro Del Piero.

Additional information

Communicated by Andreas Öchsner.

Dedicated to the memory of Bernard D. Coleman

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Del Piero, G. A mechanical model for heat conduction. Continuum Mech. Thermodyn. 32, 1159–1172 (2020). https://doi.org/10.1007/s00161-019-00821-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-019-00821-y

Keywords

Search

Navigation