Summary
The work develops the principle of virtual power for finite velocity fields for so-called simple materials (or first-gradient theory) without further constitutive assumptions when the body is swept out by a singular surface which is either afree singular surface (such as usual strong discontinuities of continuum mechanics) or athermodynamical singular surface (a so-called interface between phases). The formulation given on exemplary cases first shows how to systematically construct the new “internal” contact forces which exist at the discontinuity, as well as the new inertial contributions which arise from mass transfer across the singular surface and the acceleration of particles attached to it. Then it is shown how various virtual velocity fields generate all the dynamical field equations as well as transversality conditions when the description of external forces allows for them. The principle of virtual power here is so formulated that, when combined, forreal velocity fields, with the first principle of thermodynamics in global form, it yields directly the socalled energy theorem both in the bulk and at the singular surface. Then the corresponding rates of entropy production are deduced after introduction of the second principle of thermodynamics. While one does not claim to obtain here essentially new equations, the present formulation of the principle of virtual power paves the way for useful complex extensions which are difficult to deal with through other avenues (e.g., electromagnetic continua with “junctions” such as piezoelectric semiconductors).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Germain, P.: La méthode des puissances virtuelles en mécanique des millieux continus: théorie du second gradient. J. Mécanique12, 235–274 (1974).
Germain, P.: The method of virtual power in continuum mechanics, part 2: Microstructure. S.I.A.M. J. Appl. Math.25, 556–575 (1973).
Maugin, G. A.: The method of virtual power in continuum mechanics. Application to coupled fields. Acta Mechanics35, 1–70 (1980).
Barrere, M., Prud'homme, R.: Equations fondamentales de l'Aérothermochimie. Paris: Masson 1973.
Delhaye, J. M.: Jump conditions and entropy sources in two-phase systems—local instant formulation. Int. J. Multiphase Flow1, 395–407 (1974).
Bedeaux, D., Albano, A. M., Mazur, P.: Boundary conditions and non-equilibrium thermodynamics. Physica82A, 438–462 (1975).
Gatignol, R.: Lecture notes: Interfaces fluide-fluide Université Pierre et Marie Curie (Paris VI), Mécanique théorique, Paris (1984, unpublished).
Gogosov, V. V., Naletova, V. A., Bin, Chung Za Shaposhnikova G. A.: Conservation laws for the mass, momentum and energy and a phase interface for true and excess surface parameters. Izv. Akad. Nauk SSSR. Mekh. Zhid., Gaza (English translation)6, 923–930 (1982).
Wolf, P. A., Albano, A. M.: Non equilibrium thermodynamics of interfaces including electromagnetic effect. Physica98A, 491–508 (1979).
Daher, N., Maugin, G. A.: Modéle phénoménologique de semiconducteurs piézoélectriques. C.R. Acad. Sci. Paris299II, 999–1002 (1984).
Collet, B., Maugin, G. A.: Couplage magnétoélastique de surface dans les matériaux ferromagnétiques. C.R. Acad. Sci. Paris280A, 1641–1644 (1975).
Eringen, A. C.: Mechanics of continua. New York: J. Wiley 1967.
Gatignol, R.: Liquid-vapor interface conditions. (Submitted for publication, 1985).
Zielinska, B. J. A., Bedeaux, D.: A hydrodynamic theory for fluctuations around equilibrium of a liquid-vapour interface. Physica112A, 265–286 (1982).
Maugin, G. A., Eringen, A. C.: On the equations of the electrodynamics of deformable bodies of finite extent. J. Mécanique16, 101–147 (1977).
Katchanov, L.: Eléments de la théorie de la plasticté (French translation from the russian edition), Sec. 64. Moscow: Edition MIR 1975.
Author information
Authors and Affiliations
Additional information
With 2 Figures
Rights and permissions
About this article
Cite this article
Daher, N., Maugin, G.A. The method of virtual power in continuum mechanics application to media presenting singular surfaces and interfaces. Acta Mechanica 60, 217–240 (1986). https://doi.org/10.1007/BF01176354
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01176354