Abstract
We revisit the notion of electromagnetomechanical interactions in a general continuum, recalling first the microscopic approach in the manner of Lorentz, and then passing to a continuum via some average. The source terms thus obtained may be carried in a direct approach to the balance laws of continuum thermomechanics. A second approach consists in exploiting a generalized form of the principle of virtual power. Finally, using a variational principle of the Hamiltonian-Lagrangian form is also mentioned, although necessarily limited to nondissipative processes but presenting also some advantages (in computations, studies of stability). The common features and contrasted ones between these approaches are emphasized. It is believed that this paves the way for further modelling of this type of complex continua while accounting for the most recent works in the field as well as the author’s past works.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Maxwell J. C., (1873) A Treatise on Electricity and Magnetism, Oxford, Reprint Dover, New York (1954)
Heaviside O., (1892) Electrical papers (2 volumes, collected works), The Electrician, Macmilan, London, Reprint, Chelsea (1970)
Truesdell C.A. and Toupin R.A., The Classical Field theory, Handbuch der Physik, Vol. III/1, Ed. S. Flügge, Springer Verlag, Berlin (1960)
Mindlin R.D., Elasticity, piezoelectricity and crystal lattice dynamics, J. Elasticity, 2, 217–282 (1972)
Eringen A.C., Mechanics of Continua (revised and enlarged edition; see new Chapter 10), Krieger, New York (1980)
Brown W.F., Magnetoelastic interactions, Springer-Verlag, New York (1966)
Tiersten H.F., A development of the equations of electromagnetism in material continua, Springer-Verlag, New York (1990)
Nelson D.F., Electric, optic and acoustic interactions in dielectrics, J. Wiley-Interscience, New York (1979)
Maugin G.A. and Eringen A.C., On the equations of the electrodynamics of deformable bodies of finite extent, J. Mécanique (Paris), 16, 1–47 (1977)
Eringen A.C. and Maugin G.A., Electrodynamics of Continua, Vol. I and II, Springer-Verlag, New York (1990)
Maugin G.A., Continuum Mechanics of Electromagnetic Solids, North-Holland, Amsterdam (1988)
Lorentz H.A., (1900) Theory of Electrons, Reprint, Dover, New York (1952)
De Groot S.R. and Suttorp L.G., Foundations of electrodynamics, North-Holland, Amsterdam (1972)
Maugin G.A., Thermodynamics of Nonlinear Dissipative Phenomena, World Scientific, Singapore (1999)
Kovetz A., Electromagnetic Theory, Oxford University Press, Oxford (2000)
Ericksen J.L., On formulating and assessing continuum theories of electromagnetic fields in elastic materials, J. Elasticity, 87, 95–108 (2007)
Ericksen J.L., Magnetizable and polarizable elastic materials, Meth. Mech. Solids, 13, 38–54 (2008)
Steigmann D.J., Equilibrium theory for magnetoelastomers and magnetoelastic membranes, Int. J. Non-linear Mech., 39, 1193–1216 (2004)
Dorfmann A. and Ogden R.W., Nonlinear electroelasticity, Acta Mechanica, 174, 167–183 (2005)
Dorfmann A. and Ogden R.W., Nonlinear electroelastic deformations, J. Elasticity, 88, DOI: 10:2007/s10659-005-9028-y (2006)
Dorfmann A. and Ogden R.W., Nonlinear magnetoelastic deformations of elastomers, Acta Mechanica, 167, 13–28 (2003)
Dorfmann A. and Ogden R.W., Nonlinear magnetoelastic deformations, Quart. J. Mech. Appl. Math., 57, 599–622 (2004)
Kankanala S.V. and Triantafyllidis N., Magnetoelastic buckling of rectangular block in plane strain, J. Mech. Phys. Solids, 56, 1147–1169 (2008)
Ottenio M., Destrade M. and Ogden R.W., Incremental magnetoelastic deformations with application to surface instability, J. Elasticity, 90, 19–42 (2008)
Livens G.H., The theory of electricity, 2nd Edition, Cambridge University Press, London (1962)
Dixon R.C. and Eringen A.C., A dynamical theory of polar elastic dielectrics I, II, Int. J. Engng. Sci., 3, 359–377, 379–398 (1964)
Collet B. and Maugin G.A., On the electrodynamics of continua with interactions (in French), C. R. Acad. Sci. Paris, 279B, 379–382 (1974)
Maugin G.A., Material inhomogeneities in elasticity, Chapman & Hall, London (1993)
Trimarco C. and Maugin G.A.,Material mechanics of electromagnetic solids, in: Configurational Mechanics of Materials, (Eds.) Kienzler R. and Maugin G.A., pp. 129–171, Springer, Wien (2001)
Maugin G.A., On the foundations of the electrodynamics of deformable media with interactions, Lett. Appl. Engng. Sci., 4, 3–17 (1976)
Maugin G.A., The principle of virtual power in continuum mechanics. Application to coupled fields, Acta Mechanica, 35, 1–70 (1980)
Toupin R.A., The elastic dielectric, J. Rat. Mech. Anal., 5, 849–916 (1956)
Tiersten H.F., Coupled magnetomechanical equations for magnetically saturated insulators, J. Math. Phys., 5, 1298–1318 (1964)
Daher N. and Maugin G.A., Virtual Power and Thermodynamics for Electromagnetic Continua with Interfaces, J. Math. Phys., 27, 3022–3035 (1986)
Daher N. and 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)
Daher N. and Maugin G.A., Nonlinear Electroacoustic Equations in Semi-conductors with Interfaces (Relation Between the Macroscopic and Quasi-microscopic Descriptions), Int. J. Engng. Sci., 26, 37–58 (1988)
Rajagopal R.K. and Ruzicka M., Mathematical modelling of electrorheological materials, Continuum Mech. & Thermodyn., 13, 59–78 (2001)
Drouot R. and Racineux G., A continuum modelling for the reversible behavior of electrorheological media, Intern. J. Appl. Electromagn. Mech., 22, 177–187 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Maugin, G.A. On modelling electromagnetomechanical interactions in deformable solids. Int J Adv Eng Sci Appl Math 1, 25–32 (2009). https://doi.org/10.1007/s12572-009-0002-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12572-009-0002-y