Exegesis of the Introduction and Sect. I from Fundamentals of the Mechanics of Continua by E. Hellinger

The fundamental review article DIE ALLGEMEINEN ANSATZE DER MECHANIK DER KONTINUA in the Encyklopadie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Bd. IV-4, Hft. 5 (1913) by Ernst Hellinger has not been translated into English so far. We believe that such a circumstance is really deprecative, as the insight reached by Hellinger in the mathematical structure of continuum mechanics seems, in some aspects, unsurpassed even nowadays. Hellinger's scientific manuscripts do not fill more than one and half boxes in library storage [ 2], but their impact on mathematics and mechanical sciences is profound. Indeed, the Hellinger-Reissner variational principle is still a fundamental tool in theoretical and numerical mechanics. The intent of this paper is threefold: i) to allow to those who cannot understand German to enjoy the reading of a crystal-clear and still topical article whose content has some enlightening parts, ii) to show that only one century ago the principle of virtual work (or virtual velocities) was regarded as the central principle in continuum mechanics and that Hellinger did forecast already then the main lines of its development, iii) to discuss some technical and conceptual aspects of the variational principles in continuum mechanics which some authors consider still controversial. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim