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Edelen’s dissipation potentials and the visco-plasticity of particulate media

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An Erratum to this article was published on 11 November 2014

Abstract

Following is an elaboration on D. G. B. Edelen’s (1972–1973) nonlinear generalization of the classical Rayleigh-Onsager dissipation potentials and the implications for the models of viscoplasticity. A brief derivation is given via standard vector calculus of Edelen’s potentials and the associated non-dissipative or “gyroscopic” forces and fluxes. It is also shown that certain extensions of Edelen’s formulae can be obtained by means of a recently proposed source-flux relation or “inverse divergence,” a generalization of the classical Gauss-Maxwell construct. The Legendre–Fenchel duality of Edelen’s potentials is explored, with important consequences for rate-independent friction or plasticity. The use of dissipation potentials serves to facilitate the development of viscoplastic constitutive equations, a point illustrated here by the special cases of Stokesian fluid-particle suspensions and granular media. In particular, we consider inhomogeneous systems with particle migration coupled to gradients in particle concentration, strain rate, and fabric. Employing a mixture-theoretic treatment of Stokesian suspensions, one is able to identify particle stress as the work conjugate of the global deformation of the particle phase. However, in contrast to past treatments, this stress is not assumed to be a privileged driving force for particle migration. A comparison is made with models based on extremal dissipation or entropy production. It is shown that such models yield the correct dissipative components of force or flux but generally fail to capture certain non-dissipative, but mechanically relevant components. The significance of Edelen’s gyroscopic forces and their relation to reactive constraints or other reversible couplings is touched upon. When gyroscopic terms are absent, one obtains a class of strongly dissipative or hyperdissipative materials whose quasi-static mechanics are governed by variational principles based on dissipation potential. This provides an interesting analog to elastostatic variational principles based on strain energy for hyperelastic materials and to the associated material instabilities arising from loss of convexity.

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References

  1. Acosta G., Duràn R., Muschietti M.: Solutions of the divergence operator on John domains. Adv. Math. 206, 373–401 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ball J.M.: Singularities and computation of minimizers for variational problems.. In: DeVore, R.A., Iserles, A., Endre, S. (eds) Foundations of Computational Mathematics, Volume 284 of London Mathematical Society Lecture Notes, pp. 1–19. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  3. Bataille J., Edelen D., Kestin J.: Nonequilibrium thermodynamics of the nonlinear equations of chemical kinetics. J. Non Equil. Thermo. 3, 153–68 (1978)

    Google Scholar 

  4. Bataille J., Edelen D., Kestin J.: On the structuring of thermodynamic fluxes: a direct implementation of the dissipation inequality. Int. J. Eng. Sci. 17, 563–72 (1979)

    Google Scholar 

  5. Besson J.: Continuum models of ductile fracture: a review. Int. J. Damage Mech. 19, 3–52 (2010)

    Article  Google Scholar 

  6. Collins I., Houlsby G.: Application of thermomechanical principles to the modelling of geotechnical materials. Proc. Roy. Soc. Lond. A. 453, 1975–2001 (1997)

    Article  MATH  Google Scholar 

  7. Cowin S.C.: The relationship between the elasticity tensor and the fabric tensor. Mech. Mater. 4, 137–47 (1985)

    Google Scholar 

  8. Edelen D.G.B.: A nonlinear Onsager theory of irreversibility. Int. J. Eng. Sci. 10, 481–90 (1972)

    Google Scholar 

  9. Edelen D.G.B.: On the existence of symmetry relations and dissipation potentials. Arch. Ration. Mech. Anal. 51, 218–27 (1973)

    Google Scholar 

  10. Edelen D.G.B.: College Station Lectures on Thermodynamics. Texas A & M University, College Station (1993)

    Google Scholar 

  11. Edelen, D.G.B.: Applied exterior calculus. Dover Publications Inc., Mineola, NY, revised edition, (2005)

  12. Edelen D.G.B.: Properties of an elementary class of fluids with nondissipative viscous stresses. Int. J. Eng. Sci. 15, 727–31 (1977)

    Google Scholar 

  13. Effros E.G.: A matrix convexity approach to some celebrated quantum inequalities. PNAS 106, 1006–08 (2009)

    Google Scholar 

  14. Eringen A.C.: Microcontinuum Field Theories, Volume I: Foundations and Solids. Springer, New York (1999)

    Book  Google Scholar 

  15. Fang Z., Mammoli A., Brady J., et al.: Flow-aligned tensor models for suspension flows. Int. J. Multiph. Flow. 28, 137–66 (2002)

    Google Scholar 

  16. Freudenthal A., Geiringer H.: The mathematical theories of the inelastic continuum. In: Flügge, H. (eds) , Elasticity and Plasticity, Volume VI of Handbuch der Physik, pp. 229–432. Springer, Berlin (1958)

    Google Scholar 

  17. Gao D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods, and Applications, Volume 39. Kluwer Academic Publishers, Dordrect (2000)

    Book  Google Scholar 

  18. Goddard J.D.: Material instability in complex fluids. Ann. Rev. Fluid Mech. 35, 113–33 (2003)

    Google Scholar 

  19. Goddard, J.D.: A weakly nonlocal anisotropic fluid model for inhomogeneous Stokesian suspensions. Phys. Fluids, 20, 040601/1–01/16, (2008)

  20. Goddard J.D.: Parametric hypoplasticity as continuum model for granular media: from Stokesium to Mohr-Coulombium and beyond. Gran. Mat. 12, 145–50 (2010)

    Google Scholar 

  21. Goddard J.D.: A note on Eringen’s moment balances. Int. J. Eng. Sci. 49, 1486–93 (2011)

    Google Scholar 

  22. Goddard J.D.: On the thermoelectricity of W. Thomson: towards a theory of thermoelastic conductors. J. Elast. 104, 267–80 (2011)

    Google Scholar 

  23. Green A.E., Naghdi P.M., Trapp J.A.: Thermodynamics of a continuum with internal constraints. Int. J. Eng. Sci. 8, 891–908 (1970)

    Article  MATH  Google Scholar 

  24. Gurtin M.E.: Continuum thermodynamics. In: Nemat-Nasser, S. (eds) Mechanics Today, pp. 168–213. Pergamon, New York (1972)

    Google Scholar 

  25. Hill R.: A variational principle of maximum plastic work in classical plasticity. QJMAM 1, 18–28 (1948)

    Article  MATH  Google Scholar 

  26. Hill R.: The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950)

    MATH  Google Scholar 

  27. Hill R., Rice J.: Elastic potentials and the structure of inelastic constitutive laws. SIAM J. Appl. Math. 25, 448–61 (1973)

    Google Scholar 

  28. Johnson M.W.: On variational principles for non-Newtonian fluids. Trans. Soc. Rheol. 5, 9–21 (1961)

    Article  Google Scholar 

  29. Keller J., Rubenfeld L., Molyneux J.: Extremum principles for slow viscous flows with applications to suspensions. J. Fluid Mech. 30, 97–125 (1967)

    Article  MATH  Google Scholar 

  30. Lehoucq R., Silling S.: Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56, 1566–1577 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lighthill M.J.: Introduction to Fourier analysis and Generalised Functions. Cambridge Monographs on Mechanics and Applied Mathematics. University Press, Cambridge (1958)

    Google Scholar 

  32. Lippmann H.: Eine Cosserat-Theorie des plastischen Fließens. Acta Mech. 8, 255–284 (1969)

    Article  MATH  Google Scholar 

  33. Lippmann H.: Extremum and Variational Principles in Mechanics, Volume 54 of International Centre for Mechanical Sciences Courses and Lectures. Springer, New York (1972)

    Google Scholar 

  34. Maréchal P.: On a functional operation generating convex functions, part 2: Algebraic properties. J. Optim. Theor. Appl. 126, 357–66 (2005)

    Article  MATH  Google Scholar 

  35. Maugin G.A.: The Thermomechanics of Plasticity and Fracture. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  36. Mohan S., Rao K., Nott P.: A frictional Cosserat model for the slow shearing of granular materials. J. Fluid Mech. 457, 377–409 (2002)

    MATH  MathSciNet  Google Scholar 

  37. Moreau J.J.: Sur les lois de frottement, de plasticité et de viscosité. CR Acad. Sci. 271, 608–11 (1970)

    Google Scholar 

  38. Nemat-Nasser S.: Plasticity: A Treatise on Finite Deformation of Heterogeneous Inelastic Materials. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  39. Ostoja-Starzewski M., Zubelewicz A.: Powerless fluxes and forces, and change of scale in irreversible thermodynamics. J. Phys. A. 44, 335002 (2011)

    Article  MathSciNet  Google Scholar 

  40. Panagiotopoulos P.D.: Non-convex superpotentials in the sense of F.H. Clarke and applications. Mech. Res. Commun. 8, 335–40 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  41. Prager W.: On ideal locking materials. Trans. Soc. Rheol. 1, 169–75 (1957)

    Article  MATH  Google Scholar 

  42. Rajagopal K., Tao L.: Mechanics of Mixtures, vol. 35 of Series on Advances in Mathematics for Applied Sciences. World Scientific, River Edge (1995)

    Google Scholar 

  43. Rajagopal K.R., Srinivasa A.R.: A thermodynamic framework for rate type fluid models. J. Non-Newtonian Fluid Mech. 88, 207–27 (2000)

    Article  MATH  Google Scholar 

  44. Regirer S.A., Rutkevich I.M.: Certain singularities of the hydrodynamic equations of non-Newtonian media. J. Appl. Math. Mech. 32, 62–66 (1968)

    Article  Google Scholar 

  45. Segev R., De Botton G.: On the consistency conditions for force systems. Int. J. Non. Lin. Mech. 26, 47–59 (1991)

    Article  MathSciNet  Google Scholar 

  46. Sun J., Sundaresan S.: A constitutive model with microstructure evolution for flow of rate-independent granular materials. J. Fluid Mech. 682, 590–616 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  47. Wikipedia.: Convex Conjugate—Wikipedia the free encyclopedia, 2011. http://en.wikipedia.org/wiki/Convex_conjugate

  48. Yapici K., Powell R.L., Phillips R.J.: Particle migration and suspension structure in steady and oscillatory plane Poiseuille flow. Phys. Fluids 21, 053302–16 (2009)

    Article  Google Scholar 

  49. Ziegler, H.: Some extremum principles in irreversible thermodynamics with application to continuum mechanics. In Progress in Solid Mechanics, pp. 93–192. J. Wiley; North-Holland Pub. Co., New York; Amsterdam, (1963)

  50. Ziegler H.: Discussion of some objections to thermomechanical orthogonality. Arch. Appl. Mech. 50, 149–64 (1981)

    MATH  Google Scholar 

  51. Ziegler H.: An Introduction to Thermomechanics, volume 21 of North-Holland Series in Applied Mathematics and Mechanics. Elsevier Science, Amsterdam (1983)

    Google Scholar 

  52. Ziegler H., Wehrli C.: On a principle of maximal rate of entropy production. J. Non. Equil. Thermo. 12, 229–44 (1987)

    Google Scholar 

  53. Ziegler H. , Yu L.: Incompressible Reiner-Rivlin fluids obeying the orthogonality condition. Arch. Appl. Mech. 41, 89–99 (1972)

    MATH  Google Scholar 

Download references

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  1. Department of Mechanical and Aerospace Engineering, University of California, 9500 Gilman Drive, La Jolla, CA, 92093-0411, USA

    J. D. Goddard

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Goddard, J.D. Edelen’s dissipation potentials and the visco-plasticity of particulate media. Acta Mech 225, 2239–2259 (2014). https://doi.org/10.1007/s00707-014-1123-3

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