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 complementary variational condition. Primal, complementary and mixed variational inequalities leading to computational methods in heat-conduction boundary-value problems are briefly discussed.
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References
Cauchy, A.L.: Reserches sur l’équilibre et le movement intérieur des corps solides ou fluides, élastiques ou non élastiques. Bull. Soc. Philomath. 9–13 (1823)
Cauchy A.L.: De la pression ou tension dans un corps solide. Exerc. Math. 2, 42–56 (1827)
Cauchy A.L.: Sur les équations qui expriment les conditions d’équilibre ou les lois du movement intérieur d’un corps solide, élastiques ou non élastiques. Exerc. Math. 3, 160–187 (1828)
Noll W.: The foundations of classical mechanics in the light of recent advances in continuum mechanics. In: Suppes, P. (eds) The Axiomatic Method, with Special Reference to Geometry and Physics, pp. 266–281. North-Holland, Amsterdam (1959)
Gurtin M.E., Martins L.C.: Cauchy’s theorem in classical physics. Arch. Ration. Mech. Anal. 60, 305–324 (1976)
Šilhavý M.: Cauchy’s stress theorem for stresses represented by measures. Continuum Mech. Thermodyn. 20, 75–96 (2008)
Truesdell C., Toupin R.A.: The Classical Field Theories, Handbuch der Physik. Springer, Berlin (1960)
Piola G.: La meccanica dei corpi naturalmente estesi trattata col calcolo delle variazioni. Opuscoli Matematici Fisici Diversi Autori 201–236 (1833)
Podio Guidugli P.: A virtual power format for thermomechanics. Continuum Mech. Thermodyn. 20, 479–487 (2009)
Romano, G.: Scienza delle Costruzioni, Tomo I. Hevelius, Benevento, (2003). wpage.unina.it/romano
Romano G., Diaco M. : A functional framework for applied continuum mechanics. In: Fergola, P (eds) New Trends in Mathematical Physics, pp. 193–204. World Scientific, Singapore (2004)
Luenberger D.: Optimization by vector space methods. Wiley, New York (1968)
Euler L.: Principia motus fluidorum. Novi Comm. Acad. Sci. Petrop. 6, 271–311 (1761)
Romano, G.: Continuum Mechanics on Manifolds. Lecture Notes, University of Naples Federico II, Naples, 2008. wpage.unina.it/romano
Fichera G.: In: Flgge, S. (eds) Handbuch der Physik, Vol. VI/a, Springer, New York (1972)
Duvaut G., Lions J.L.: Inequalities in Mechanics and Physics. Springer, New York (1976)
Romano, G.: Theory of structures, Part I: elements of linear analysis. Lecture Notes, University of Naples Federico II, Naples, Italy, 2000. wpage.unina.it/romano (in Italian)
Caratheodory C.: Untersuchungen über die grundlagen der thermodynamik. Math. Ann. 67, 355–386 (1909)
Fermi E.: Thermodynamics. Dover Publications, New York (1936)
Truesdell C.: Rational Thermodynamics. 2nd edn. Springer, New York (1984)
Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)
Peetre J.: Another approach to elliptic boundary problems. Commun. Pure Appl. Math. 14, 711–731 (1961)
Tartar L.: Sur un lemme d’équivalence utilisé en Analyse Numérique. Calcolo XXIV(II), 129–140 (1987)
Romano G.: On the necessity of Korn’s inequality. In: O’Donoghue, P.E., Flavin, J.N. (eds) Symposium on Trends in Applications of Mathematics to Mechanics, pp. 166–171. Elsevier, Galway (2000)
Aubin J.P.: Approximation of Elliptic Boundary-Value Problems. Wiley, New York (1972)
Yosida K.: Functional Analysis, 4th edn. Springer, New York (1974)
Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)
Fourier J.B.J.: Théorie Analytique de la Chaleur. Firmin-Didot, Parigi (1822)
Romano G.: New results in subdifferential calculus with applications to convex optimization. Appl. Math. Optim. 32, 213–234 (1995)
Panagiotopouls P.D.: Inequality Problems in Mechanics and Applications with Applications to Convex Optimization. Birkhäuser, Boston (1985)
Šilhavý M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin (1997)
Romano, G., Barretta, R.: On the Variational Formulation of Balance Laws. University of Naples Federico II, Naples (2008) (Preprint)
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Romano, G., Diaco, M. & Barretta, R. Variational Formulation of the First Principle of Continuum Thermodynamics. Continuum Mech. Thermodyn. 22, 177–187 (2010). https://doi.org/10.1007/s00161-009-0119-z
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DOI: https://doi.org/10.1007/s00161-009-0119-z