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Advanced Methods of Continuum Mechanics for Materials and Structures pp 547–556Cite as

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

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

Abstract

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.

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References

  • Aifantis, E.C.: Update on a class of gradient theories. Mech. Mater. 35(3–6), 259–280 (2003)

    Article  Google Scholar 

  • Altenbach, H., Eremeyev, V.A. (eds.): Generalized Continua: From the Theory to Engineering Applications. CISM Courses and Lectures. Springer, Wien (2013)

    Google Scholar 

  • Altenbach, H., Maugin, G.A., Erofeev, V. (eds.): Mechanics of Generalized Continua, Advanced Structured Materials, vol. 7. Springer, Berlin (2011)

    Google Scholar 

  • Altenbach, H., Forest, S., Krivtsov, A. (eds.): Generalized Continua as Models for Materials, Advanced Structured Materials, vol. 22. Springer, Berlin (2013)

    Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Brenner, H.: Navier-Stokes revisited. Phys. A: Stat. Mech. Appl. 349(1–2), 60–132 (2005)

    Article  MathSciNet  Google Scholar 

  • Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258 (1958)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Capriz, G.: Continua with Microstructure. Springer, New York (1989)

    Book  MATH  Google Scholar 

  • de Gennes, P.G., Brochard-Wyart, F., Quéré, D.: Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York (2004)

    Book  MATH  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • 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)

    MATH  Google Scholar 

  • dell’Isola, F., Seppecher, P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • dell’Isola, F., Steigmann, D.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118(1), 113–125 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

  • Dunn, J.E., Serrin, J.: On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88(2), 95–133 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  • Eremeyev, V.A., Altenbach, H.: Equilibrium of a second-gradient fluid and an elastic solid with surface stresses. Meccanica 49(11), 2635–2643 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer-Briefs in Applied Sciences and Technologies. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  • Eringen, A.C.: Microcontinuum Field Theory I. Foundations and Solids. Springer, New York (1999)

    Book  MATH  Google Scholar 

  • Eringen, A.C.: Microcontinuum Field Theory II. Fluent Media. Springer, New York (2001)

    MATH  Google Scholar 

  • Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

  • Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)

    Article  MATH  Google Scholar 

  • Forest, S.: Some links between Cosserat, strain gradient crystal plasticity and the statistical theory of dislocations. Philos. Mag. 88(30–32), 3549–3563 (2008)

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Germain, P.: The method of virtual power in continuum mechanics. part 2: Microstructure. SIAM J. Appl. Math. 25(3), 556–575 (1973b)

    Article  MathSciNet  MATH  Google Scholar 

  • Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113–147 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Heida, M., Málek, J.: On compressible Korteweg fluid-like materials. Int. J. Eng. Sci. 48(11), 1313–1324 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010)

    Book  MATH  Google Scholar 

  • 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)

    MATH  Google Scholar 

  • Maugin, G.A., Muschik, W.: Thermodynamics with internal variables. Part II. Applications. J. Non-Equilib. Thermodyn. 19(3), 250–289 (1994)

    MATH  Google Scholar 

  • Mühlhaus, H.B., Aifantis, E.C.: A variational principle for gradient plasticity. Int. J. Solids Struct. 28(7), 845–857 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Pietraszkiewicz, W., Eremeyev, V., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM 87(2), 150–159 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Rowlinson, J.S., Widom, B.: Molecular Theory of Capillarity. Dover, New York (2003)

    Google Scholar 

  • Sedov, L.I.: Models of continuous media with internal degrees of freedom. J. Appl. Math. Mech. 32(5), 803–819 (1968)

    Article  MATH  Google Scholar 

  • 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)

    MathSciNet  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Seppecher, P.: Les fluides de Cahn-Hilliard. Mémoire d’habilitation à diriger des recherches, Université du Sud Toulon (1996)

    MATH  Google Scholar 

  • 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)

    Chapter  Google Scholar 

  • Truesdell, C.: The Elements of Continuum Mechanics. Springer, New York (1966)

    MATH  Google Scholar 

  • Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The Faculty of Mechanical Engineering, Rzeszów University of Technology, al. Powstańców Warszawy 8, 35-959, Rzeszów, Poland

    Victor A. Eremeyev

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  1. Victor A. Eremeyev
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Correspondence to Victor A. Eremeyev .

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  1. Institute of Mechanics IFME, Otto von Guericke University, Magdeburg, Sachsen-Anhalt, Germany

    Konstantin Naumenko

  2. Otto von Guericke University, Institute of Mechanics, Magdeburg, Germany

    Marcus Aßmus

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Eremeyev, V.A. (2016). On Equilibrium of a Second-Gradient Fluid Near Edges and Corner Points. In: Naumenko, K., Aßmus, M. (eds) Advanced Methods of Continuum Mechanics for Materials and Structures. Advanced Structured Materials, vol 60. Springer, Singapore. https://doi.org/10.1007/978-981-10-0959-4_29

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  • DOI: https://doi.org/10.1007/978-981-10-0959-4_29

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