Continuum Mechanics and Thermodynamics

, Volume 22, Issue 3, pp 177–187 | Cite as

Variational Formulation of the First Principle of Continuum Thermodynamics

Original Article


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.


Continuum Thermodynamics Lagrange multipliers Virtual temperatures Heat flow 


45.20.dh 46.15.Cc 44.10.+i 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Giovanni Romano
    • 1
  • Marina Diaco
    • 1
  • Raffaele Barretta
    • 1
  1. 1.Department of Structural EngineeringUniversity of Naples Federico IINaplesItaly

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