Variational formulation of the first principle of continuum thermodynamics

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.