Abstract
An algebraic characterization of fluidity applicable to second-gradient materials is argued, individuating a collection of deformations from a reference placement that entail no pointwise stress-power expenditure. For simplicity, the characterization in question is developed in the context of elastic materials, within which general representations for the stress response, both Cauchy-like and Piola-like, of elastic second-gradient fluids are derived.
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Degiovanni M., Marzocchi A., Musesti A.: Edge-force densities and second-order powers. Ann. Mat. Pura. Appl. 1, 81–103 (2006)
Dell’Isola F., Seppecher P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)
Dunn J.E., Serrin J.: On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88(2), 95–133 (1985)
Forte S., Vianello M.: On surfaces 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.: Thermomechanics of the interface between a body and its environment. Continuum Mech. Thermodyn. 19(5), 253–271 (2007)
Fried, E., Gurtin, M.E.: Turbulent kinetic energy and a possible hierarchy of length scales in a generalization of the Navier–Stokes alpha theory. Phys. Rev. E. Stat. Nonlin. Soft Matter Phys. 75(5, Part 2), 056306 (2007)
Fried E., Gurtin M.E.: A continuum mechanical theory for turbulence: a generalized Navier-Stokes-alpha equation with boundary conditions. Theor. Comput. Fluid Dyn. 22(6), 433–470 (2008)
Germain P.: La méthode des puissances virtuelles en mécanique des milieux continus. Première partie: théorie du second gradient. J. Méc. 12, 235–274 (1973)
Germain P.: The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)
Gurtin M.E.: An Introduction to Continuum Mechanics. Mathematics in Science and Engineering. Academic Press, New York (1981)
Gurtin M.E., Vianello M., Williams W.O.: On fluids of grade n. Meccanica 21(4), 179–183 (1986)
Korteweg D.J.: Sur la forme que prennent les equations du mouvement des fluides si l’on tient compte des forces capillaires. Arch. Neerl. Sci. Ex. Nat. 6, 1–24 (1901)
Noll W., Virga E.G.: On edge interactions and surface tension. Arch. Ration. Mech. Anal. 111(1), 1–31 (1990)
Podio-Guidugli P.: Contact interactions, stress, and material symmetry, for nonsimple elastic materials. Theor. Appl. Mech. 28–29, 261–276 (2002)
Podio-Guidugli, P.: On the aggregation state of simple materials. In: Šilhavý, M. (ed.) Mathematical Modeling of Bodies With Complicated Bulk and Boundary Behavior Quaderni di Matematica, vol. 20, pp. 159–168. Aracne (2007)
Podio-Guidugli P., Vianello M.: Hypertractions and hyperstresses convey the same mechanical information. Continuum Mech. Thermodyn. 22, 163–176 (2010)
Testa V., Vianello M.: The symmetry group of gradient sensitive fluids. Int. J. Nonlinear Mech. 40(5), 621–631 (2005)
Toupin R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
Toupin R.A.: Theories of elasticity with couple-stresses. Arch. Ration. Mech. Anal. 17, 85–112 (1964)
Truesdell C.A., Noll W.: The Non-Linear Field Theories of Mechanics. Springer, Berlin (2010)
Vianello M.: On the equilibrium theory of second grade fluids. Arch. Mech. (Arch. Mech. Stos.) 41, 641–649 (1989)
Warner F.W.: Foundations of Differentiable Manifolds and Lie Groups. Scott Foresman and Company, Glenview (1971)
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Communicated by Francesco dell'Isola and Samuel Forest.
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Podio-Guidugli, P., Vianello, M. On a stress-power-based characterization of second-gradient elastic fluids. Continuum Mech. Thermodyn. 25, 399–421 (2013). https://doi.org/10.1007/s00161-012-0267-4
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DOI: https://doi.org/10.1007/s00161-012-0267-4