Continuum Mech. Thermodyn. (2010) 22:177–187 DOI 10.1007/s00161-009-0119-z ORIGINAL ARTICLE Giovanni Romano · Marina Diaco · Raffaele Barretta Variational Formulation of the First Principle of Continuum Thermodynamics Received: 18 October 2008 / Accepted: 24 September 2009 / Published online: 24 October 2009 © Springer-Verlag 2009 Abstract The First Principle of Continuum Thermodynamics is formulated as a variational condition whose test fields are piecewise constant virtual temperatures. Lagrange multipliers theorem is applied to relax the constraint of piecewise constancy of test fields. This provides the existence of square summable vector fields of heat flow through the body fulfilling a virtual thermal work principle, analogous to the virtual work principle in Mechanics. The issue of compatibility of thermal gradients is dealt with and expressed by the complemen- tary variational condition. Primal, complementary and mixed variational inequalities leading to computational methods in heat-conduction boundary-value problems are briefly discussed. Keywords Continuum Thermodynamics · Lagrange multipliers · Virtual temperatures · Heat flow PACS 45.20.dh · 46.15.Cc · 44.10.+i 1 Introduction Duality is a basic concept in Mathematical Physics, and the master dual objects in Continuum Mechanics are velocity fields and force systems which interplay in the axiomatic formulation of dynamical equilibrium. The notion of a stress field in a continuous body in equilibrium was introduced in celebrated papers by Cauchy [1–3] and, starting with the pioneering contributions in [4, 5], many valuable advancements have been made to provide the existence result under milder regularity assumptions, as outlined in the recent article [6] and refer- ences therein. According to Truesdell and Toupin [7], it was Piola [8] who first applied Lagrange’s multiplier method to introduce the notion of stress field in a continuous body. His brilliant intuition was however not properly evaluated and the more easy-to-follow geometrical method by Cauchy has been reproduced almost without exceptions in mechanics textbooks, and in research articles, although, with the improvements and generalizations quoted above. The classical Cauchy method has been also adopted in [9] to propose a unified format for thermomechanics based on the notion of thermal displacement. To the best of our knowledge, the supremacy of Piola’s approach has not been fully claimed until quite recently [10, 11]. The motivation for this supremacy is twofold. On one hand, duality plays a basic role in Lagrange’s method so that the players coming into the scene are properly defined and detected. The resulting variational scheme is most fruitful and leads directly to basic theoretical results and to most efficient computational methods. On the other hand, Lagrange’s method for continuous problems is now a theorem. The existence of Lagrange’s multipliers may indeed be formally motivated by the orthogonality relation between the kernel of a linear operator and the image of the dual operator, when the involved linear spaces are finite dimensional. In the infinite dimensional context of Continuum Mechanics, the topological properties of the involved linear functional spaces and operators must be properly specified and then the existence proof for the multipliers can be given by relying on standard tools G. Romano (B ) · M. Diaco · R. Barretta Department of Structural Engineering, University of Naples Federico II, via Claudio 21, 80125 Naples, Italy E-mail: romano@unina.it