The solution of the heat equation subject to the specification of energy
This procedure represents an extension of Castigliano's principle for stresses, in the formulation of Reissner [1], to thermoelasticity and heat conduction. It is also noted, that Biot's [3] variational procedure, which yields the elastic equilibrium equations and the heat conduction equation as the Euler equations, is a mixed one, in the sense that the former is one of the equations obtained in applying the extended Green procedure, while the latter belongs to the equations obtained using the extended Castigliano's procedure. 6. Comparison of variational principles. Following Reissner's presentation and proof in isothermal elasticity, a comparison can be made between the values of I for functions tu , , etc. which are not solutions of SI = 0 and for functions r,-,, etc. which are determined from 81 = 0. If both W and D are positive definite quadratic forms, the conclusion is reached that in the extended variational theorem for displacements (solid and entropy) one is concerned with a minimum problem, while in the extended variational theorem for stresses one is concerned with a maximum problem. In contrast to this, Reissner's general variational theorem, as extended to thermoelasticity and heat conduction, is only a stationary-value problem.