The problem of a definition of a 'surface tension' or 'surface stress' in solid-liquid interfaces has been one of the most controversial topics in the Gibbs' theory of capillarity (the effect of adsorption on surface tension) and electrocapillarity (the effect of electric charge on surface tension) during the past several decades. Although Gibbs was careful not to introduce the definition of a 'surface stress', later, 75 years after Gibbs, the attempt to introduce the concept 'surface stress' was undertaken by Shuttleworth (1950). This famous equation continues to be used in many hundreds works. However, such definition of a surface stress as the strain derivative of the total free surface energy divided by the surface area is inconsistent with the usual definition of a stress tensor (given in continuum elasticity theory) as the strain derivative of the specific free energy. The error of this equation was shown even 15 years ago [E. M. Gutman: J. Phys.: Condens. Matter 7 (1995) L663].
All problems associated with long-term discussions (unfortunately, useless) about the surface stress are caused by the occurrence of the Shuttleworth equation that gave rise to a sequence of equations of Herring-Eriksson-Couchman-Jesser-Everet-Davidson-Gokhshtein-Frumkin-Rusanov (1950-2011). They are a chain of corollaries of the original equation given by Shuttleworth (1950) in scalar form and by Herring (1953) in tensorial form (SHE). Despite differences in derivations proposed by these authors, we show that all derivations are based on the common principal mathematical defect [E.M. Gutman: J. Phys.: Condens. Matter 22 (2010) 428001; Surf. Sci. 605 (2011) 644; Surf. Sci. 605 (2011) 1872; Surf. Sci. 605 (2011) 1923]. All existed derivations of SHE do not take into account an obvious constraint that consists in impossibility to equate a partial differential and a total differential of energy taken for different thermodynamic systems. Experimental verifications of mentioned equations are impossible excluding such for Lippmann and Gokhshtein equations, but latter is questionable.
An example: the erroneous Gokhshtein' electrocapillary equation can be obtained by combining correct Lippmann' electrocapillary equation (derived by Gibbs) and incorrect SHE, etc. We show that the direct connection between the electrode potential and surface charge density actually is not dependent on the tangential elastic strain of the flat ideally polarizable electrode, contrary to Gokhshtein and Frumkin.
Nevertheless, different authors try to prove the SHE applying Eulerian/Lagrangian coordinate transformations to the usual 2D definition of a stress tensor borrowed from the theory of elasticity. But the frames of Gibbs thermodynamics are limited to the field of small reversible deformations where the Eulerian and Lagrangian descriptions are practically indistinguishable. We analyze these anomalies in detail in order to prevent the erroneous use of the surface stress definitions.