Material Mechanics of Electromagnetic Solids
Eshelby [1–5] introduced the notion (and the naming) of Maxwell stress tensor of Elasticity having in mind the Maxwell energy-stress of electromagnetism.
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Dipartimento di Matematica Applicata « U.Dini », Universita di Pisa, Italy. 2 Laboratoire de Modelisation en Mecanique, Universite Pierre et Marie Curie, Paris, France.
1 Introduction Eshelby [1-5] introduced the notion (and the naming) of Maxwell stress tensor of Elasticity having in mind the Maxwell energy-stress of electromagnetism. In a vacuum, the electromagnetic energy-stress tensor stems almost straightforwardly from the set of Maxwell equations. These equations entail an additional equation, which has the form of a vector balance equation and in which the Maxwell stress tensor tM appears. In the electrostatic case and in a vacuum, the Maxwell stress tensor reads: (1.1)
where Eo is the electric permittivity of vacuum and E is the electric field. In a vacuum, the divergence of tM vanishes identically. We can say that the total Maxwell force acting on any closed surface of the physical space vanishes though tM itself may not vanish. This fact represents the main remarkable novelty of the Maxwell-Faraday theory of electromagnetism. Therefore, the field E (along with the electric displacement D) such as conceived by Faraday and introduced by Maxwell, introduces a physical stress tensor at a point of the 'empty' space even in the absence of electric charges in that point. Similarly the field H and B (the magnetic field and the magnetic induction, respectively) introduce an analogous tensor which adds to the electrostatic one. The Maxwell balance of forces extends to electrodynamics. In the context of electrodynamics, the divergence of the stress tensor balances the time derivative of a quantity, which is interpreted as a momentum density, having in mind the mechanical description of a continuum body. As a result, one associates a mechanical momentum to the electromagnetic fields. Once more, what may be surprising is that this momentum survives in a vacuum as the electromagnetic quantities pervade the whole physical space. R. Kienzler et al. (eds.), Configurational Mechanics of Materials © Springer-Verlag Wien 2001
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C. Trimarco and G.A. Maugin
However, severe difficulties arise in establishing univocally the contribution of the electromagnetic field to the momentum. The difficulties are strictly related to the problem of the proper expression for the Maxwell stress tensor in a material [6-11]. One of the main perplexing points for physicists has been that the Maxwell stress tensor may tum out to be not symmetric. As is known, the possible lack of symmetry for the stress tensor entails the failure ofthe balance ofthe moment of momentum. This occurrence would undermine the mechanical description. A thorough discussion of the question can be found in Nelson [6]. As the electric charge may be viewed as an inhomogeneity of the physical space, Eshelby suggested that, similarly, a material inhomogeneity or a material defect would produce a change in the elastic fields (and in the related stresses). According to Eshelby, the elastic field should vary in such a way that a balance law of the Maxwell kind would hold
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