The J-, M- and L-integrals of body charges and body forces: Maxwell meets Eshelby

Abstract

In this work, we derive the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals of body charges and point charges in electrostatics, and the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals of body forces and point forces in elasticity and we investigate their physical interpretation. Electrostatics is considered as field theory of an electrostatic scalar potential [Formula: see text] (scalar field theory) and elasticity as field theory of a displacement vector [Formula: see text] (vector field theory). One of the basic quantities appearing in the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals is the electrostatic Maxwell–Minkowski stress tensor in electrostatics and the Eshelby stress tensor in elasticity. Among others, it is shown that the [Formula: see text]-integral of body charges in electrostatics represents the electrostatic part of the Lorentz force, and the [Formula: see text]-integral of body forces in elasticity represents the Cherepanov force. The [Formula: see text]-integral between two-point sources (charges or forces) equals half the electrostatic interaction energy in electrostatics and half the elastic interaction energy in elasticity between these two-point sources. The [Formula: see text]-integral represents the configurational vector moment or torque between two body or point sources (charges or forces). Interesting mathematical and physical features are revealed through the connection of the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals with their corresponding infinitesimal generators in both theories. Several important outcomes arise from the comparison between the examined concepts in electrostatics and elasticity. Differences and similarities, that provide a deeper insight into the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals and the related quantities to them, are pointed out and discussed. The presented results show that the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals are fundamental concepts which can be applied in any field theory.