On Equilibrium of a Second-Gradient Fluid Near Edges and Corner Points

Part of the Advanced Structured Materials book series (STRUCTMAT, volume 60)


Within the framework of the model of second-gradient fluid we discuss the natural boundary conditions along edges and at corner points. As for any strain gradient model the model of second-gradient fluid demonstrates some peculiarities related with necessity of additional boundary conditions. Here using the Lagrange variational principle we derived the latter boundary conditions for various contact angles.


  1. Aifantis, E.C.: Update on a class of gradient theories. Mech. Mater. 35(3–6), 259–280 (2003)CrossRefGoogle Scholar
  2. Altenbach, H., Eremeyev, V.A. (eds.): Generalized Continua: From the Theory to Engineering Applications. CISM Courses and Lectures. Springer, Wien (2013)Google Scholar
  3. Altenbach, H., Maugin, G.A., Erofeev, V. (eds.): Mechanics of Generalized Continua, Advanced Structured Materials, vol. 7. Springer, Berlin (2011)Google Scholar
  4. Altenbach, H., Forest, S., Krivtsov, A. (eds.): Generalized Continua as Models for Materials, Advanced Structured Materials, vol. 22. Springer, Berlin (2013)Google Scholar
  5. Auffray, N., dell’Isola, F., Eremeyev, V.A., Madeo, A., Rosi, G.: Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. Brenner, H.: Navier-Stokes revisited. Phys. A: Stat. Mech. Appl. 349(1–2), 60–132 (2005)MathSciNetCrossRefGoogle Scholar
  7. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258 (1958)CrossRefGoogle Scholar
  8. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31, 688 (1959)CrossRefGoogle Scholar
  9. Capriz, G.: Continua with Microstructure. Springer, New York (1989)CrossRefMATHGoogle Scholar
  10. de Gennes, P.G., Brochard-Wyart, F., Quéré, D.: Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York (2004)CrossRefMATHGoogle Scholar
  11. dell’Isola, F., Rotoli, G.: Validity of Laplace formula and dependence of surface tension on curvature in second gradient fluids. Mech. Res. Commun. 22, 485–490 (1995)CrossRefMATHGoogle Scholar
  12. dell’Isola, F., Seppecher, P.: The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. Comptes rendus de l’Académie des sciences Série 2 321(8), 303–308 (1995)MATHGoogle Scholar
  13. dell’Isola, F., Seppecher, P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. dell’Isola, F., Steigmann, D.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118(1), 113–125 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. dell’Isola, F., Gouin, H., Rotoli, G.: Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. Eur. J. Mech. B/Fluids 15(4), 545–568 (1996)MATHGoogle Scholar
  16. dell’Isola, F., Seppecher, P., Madeo, A.: How contact interactions may depend on the shape of Cauchy cuts in \(N\)th gradient continua: approach “à la d’Alembert”. ZAMP 63, 1119–1141 (2012)MathSciNetMATHGoogle Scholar
  17. dell’Isola, F., Auffray, N., Eremeyev, V.A., Madeo, A., Placidi, L., Rosi, G.: Least action principle for second gradient continua and capillary fluids: a Lagrangian approach following Piola’s point of view. In: The Complete Works of Gabrio Piola, vol. I, pp. 606–694. Springer, Berlin (2014)Google Scholar
  18. dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060–804 (2015)Google Scholar
  19. Dunn, J.E., Serrin, J.: On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88(2), 95–133 (1985)MathSciNetCrossRefMATHGoogle Scholar
  20. Eremeyev, V.A., Altenbach, H.: Equilibrium of a second-gradient fluid and an elastic solid with surface stresses. Meccanica 49(11), 2635–2643 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer-Briefs in Applied Sciences and Technologies. Springer, Heidelberg (2013)CrossRefMATHGoogle Scholar
  22. Eringen, A.C.: Microcontinuum Field Theory I. Foundations and Solids. Springer, New York (1999)CrossRefMATHGoogle Scholar
  23. Eringen, A.C.: Microcontinuum Field Theory II. Fluent Media. Springer, New York (2001)MATHGoogle Scholar
  24. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)MATHGoogle Scholar
  25. Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)CrossRefMATHGoogle Scholar
  26. Forest, S.: Some links between Cosserat, strain gradient crystal plasticity and the statistical theory of dislocations. Philos. Mag. 88(30–32), 3549–3563 (2008)CrossRefGoogle Scholar
  27. Fried, E., Gurtin, M.E.: Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales. Arch. Ration. Mech. Anal. 182(3), 513–554 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W.: Mechanism-based strain gradient plasticity-I. Theory. J. Mech. Phys. Solids 47(6), 1239–1263 (1999)MathSciNetCrossRefMATHGoogle Scholar
  29. Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus - première partie, théorie du second gradient. Journal de Mécanique 12, 235–274 (1973a)MathSciNetMATHGoogle Scholar
  30. Germain, P.: The method of virtual power in continuum mechanics. part 2: Microstructure. SIAM J. Appl. Math. 25(3), 556–575 (1973b)MathSciNetCrossRefMATHGoogle Scholar
  31. Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113–147 (1964)MathSciNetCrossRefMATHGoogle Scholar
  32. Gurtin, M.E.: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50(1), 5–32 (2002)MathSciNetCrossRefMATHGoogle Scholar
  33. Heida, M., Málek, J.: On compressible Korteweg fluid-like materials. Int. J. Eng. Sci. 48(11), 1313–1324 (2010)MathSciNetCrossRefMATHGoogle Scholar
  34. Huang, Y., Gao, H., Nix, W.D., Hutchinson, J.W.: Mechanism-based strain gradient plasticity-II. Analysis. J. Mech. Phys. Solids 48(1), 99–128 (2000)MathSciNetCrossRefMATHGoogle Scholar
  35. Korteweg, D.J.: Sur la forme que prennent les équations des mouvements des fluides si l’on tient compte des forces capillaires par des variations de densité. Archives Néerlandaises des Sciences exactes et naturelles Sér II(6), 1–24 (1901)Google Scholar
  36. Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010)CrossRefMATHGoogle Scholar
  37. Maugin, G.A., Metrikine, A.V.: Mechanics of Generalized Continua. One Hundred Years After the Cosserats. Advances in Mechanics and Mathematics, vol. 21. Springer, New York (2010)MATHGoogle Scholar
  38. Maugin, G.A., Muschik, W.: Thermodynamics with internal variables. Part II. Applications. J. Non-Equilib. Thermodyn. 19(3), 250–289 (1994)MATHGoogle Scholar
  39. Mühlhaus, H.B., Aifantis, E.C.: A variational principle for gradient plasticity. Int. J. Solids Struct. 28(7), 845–857 (1991)MathSciNetCrossRefMATHGoogle Scholar
  40. Pietraszkiewicz, W., Eremeyev, V., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM 87(2), 150–159 (2007)MathSciNetCrossRefMATHGoogle Scholar
  41. Podio-Guidugli, P., Vianello, M.: On a stress-power-based characterization of second-gradient elastic fluids. Contin. Mech. Thermodyn. 25(2–4), 399–421 (2013)MathSciNetCrossRefGoogle Scholar
  42. Rahali, Y., Giorgio, I., Ganghoffer, J., dell’Isola, F.: Homogenization à la piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)MathSciNetCrossRefGoogle Scholar
  43. Rosi, G., Giorgio, I., Eremeyev, V.A.: Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. ZAMM 93(12), 914–927 (2013)MathSciNetCrossRefGoogle Scholar
  44. Rowlinson, J.S., Widom, B.: Molecular Theory of Capillarity. Dover, New York (2003)Google Scholar
  45. Sedov, L.I.: Models of continuous media with internal degrees of freedom. J. Appl. Math. Mech. 32(5), 803–819 (1968)CrossRefMATHGoogle Scholar
  46. Seppecher, P.: Etude des conditions aux limites en théorie du second gradient: cas de la capillarité. Comptes rendus de l’Académie des sciences Série 2, Mécanique, Physique, Chimie, Sciences de l’univers. Sciences de la Terre 309(6), 497–502 (1989a)MathSciNetGoogle Scholar
  47. Seppecher, P.: Étude des conditions aux limites en théorie du second gradient: cas de la capillarité. Comptes Rendus de l’Académie des Sciences 309, 497–502 (1989b)MathSciNetMATHGoogle Scholar
  48. Seppecher, P.: Equilibrium of a Cahn-Hilliard fluid on a wall—Influence of the wetting properties of the fluid upon the stability of a thin liquid film. Eur. J. Mech. B/Fluids 12(1), 69–84 (1993)MathSciNetMATHGoogle Scholar
  49. Seppecher, P.: Les fluides de Cahn-Hilliard. Mémoire d’habilitation à diriger des recherches, Université du Sud Toulon (1996)MATHGoogle Scholar
  50. Seppecher, P.: Second-gradient theory: application to Cahn-Hilliard fluids. In: Maugin, G.A., et al. (eds.) Continuum Thermomechanics: : The Art and Science of Modeling Matter’s Behaviour, pp. 379–388. Springer, Dordrecht (2002)CrossRefGoogle Scholar
  51. Truesdell, C.: The Elements of Continuum Mechanics. Springer, New York (1966)MATHGoogle Scholar
  52. Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
  53. van der Waals, J.D.: The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (English translation by J. S. Rowlinson). J. Stat. Phys. 20, 200–244 (1893)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.The Faculty of Mechanical EngineeringRzeszów University of TechnologyRzeszówPoland

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