Abstract
It is well known that second-gradient continuum mechanical theories allow for the appearance of concentrated stresses along the edges of piecewise smooth material surfaces, but this is not the sole example of concentrated interaction. Two additional kinds of concentrated interaction are shown to take place in some second-gradient incompressible dissipative fluids: the adherence to one-dimensional immersed bodies and the capability of sustaining concentrated external body forces. These three phenomena turn out to be distinct and independent. This feature is explicitly discussed in two benchmark problems, and the different mathematical origins of each concentrated interaction are explained.
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References
Adams R.A., Fournier J.J.F.: Sobolev spaces. Elsevier, Amsterdam (2003)
Das S.K., Choi S.U.S., Yu W., Pradeep T.: Nanofluids: Science and Technology. Wiley, Hoboken (2008)
Degiovanni M., Marzocchi A., Musesti A.: Edge-force densities and second-order powers. Ann. Mat. Pura Appl. 185(1), 81–103 (2006)
dell’Isola F., Seppecher P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)
dell’Isola, F., Seppecher, P., Madeo, A.: How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à à la D’Alembert”. Z. Angew. Math. Phys. (ZAMP). doi:10.1007/s00033-012-0197-9
Forte S., Vianello M.: On surface stresses and edge forces. Rend. Mat. Appl. 8(3), 409–426 (1988)
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)
Fried E., Gurtin M.E.: A continuum mechanical theory for turbulence: a generalized navier–stokes-α equation with boundary conditions. Theor. Comput. Fluid Dyn. 22, 433–470 (2008)
Germain P.: La méthode des puissances virtuelles en mécanique des milieux continus. I. Théorie du second gradient. J. Mécanique 12, 235–274 (1973)
Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)
Giusteri G., Marzocchi A., Musesti A.: Three-dimensional nonsimple viscous liquids dragged by one-dimensional immersed bodies. Mech. Res. Commun. 37(7), 642–646 (2010)
Giusteri G., Marzocchi A., Musesti A.: Nonsimple isotropic incompressible linear fluids surrounding one-dimensional structures. Acta Mech. 217, 191–204 (2011)
Lebedev N.N.: Special Functions and Their Applications. Dover Publications Inc., New York (1972)
Musesti A.: Isotropic linear constitutive relations for nonsimple fluids. Acta Mech. 204, 81–88 (2009)
Noll W., Virga E.G.: On edge interactions and surface tension. Arch. Ration. Mech. Anal. 111(1), 1–31 (1990)
Podio-Guidugli P., Vianello M.: Hypertractions and hyperstresses convey the same mechanical information. Contin. Mech. Thermodyn. 22(3), 163–176 (2010)
Toupin R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
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Giusteri, G.G. The multiple nature of concentrated interactions in second-gradient dissipative liquids. Z. Angew. Math. Phys. 64, 371–380 (2013). https://doi.org/10.1007/s00033-012-0229-5
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DOI: https://doi.org/10.1007/s00033-012-0229-5