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The justification of plate models by the two-energies-principle
Dietrich Braess
Ruhr-University Bochum, Germany
The solution of the Lamé equations characterizes the displacement that minimizes the induced stored energy. The associated dual problem describes the stresses that satisfy the equilibrium condition and maximize the corresponding energy. When approximate solutions of both problems are known, then the theorem of Prager and Syngy provides error bounds for both of them. The name “two-energies-principle” is now preferred when it is used for getting a posteriori error bounds.
In 1959 Morgenstern started a proof of the justification of plate models, but it was not completed. Now two changes enable a completion of the proof for the plate models by Mindlin-Reissner and Kirchhoff-Love. First, a hidden quadratic term is added to the displacement in the z-direction. Moreover, the regularity of the constructed stress tensor is determined in a sophisticated way. The error of the models depends on the thickness of the plate and on the mentioned regularity.
The details are given in a joint paper with Stefan Sauter and Christoph Schwab.
