$$\varvec{J}$$-, M- and $$\varvec{L}$$-integrals of line charges and line forces

Abstract In this work, the $$\varvec{J}$$ J -, M - and $$\varvec{L}$$ L -integrals of two line charges forces in generalized plane strain are derived framework three-dimensional (3D) electrostatics compatible linear elasticity, respectively, order to study interaction between them. The key point derivation is achieved by expressing three dimensions terms corresponding Green functions, that is, function Laplace operator tensor Navier operator, respectively. major mathematical tool used deriving $$L_3$$ 3 from method embedding or descent functions (2D) 3D. analytical expressions antiplane discussed. -integral electrostatic part Lorentz force (electrostatic force) Cherepanov elasticity. sources (charges forces) equals half energy elastic elasticity these sources, z -component configurational vector moment rotational representing total torque about -axis caused sources. An important outcome twice negative gradient -integral, leading result $${\varvec{J}}$$ for a conservative (with playing role potential energy) consequently an irrotational field. Finally, obtained being distance angle, able describe physical behaviour forces), since they represent fundamental necessary quantities, force, produced