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Three ways to derive the classical plate model

Wenbin Yu's picture

Attached is part of my lecture notes for a graduate structural mechanics. In
the notes, we derived the classical plate theory, which is also called
the Kirchhoff plate theory, in three ways: Newtonian method, variational
method, and variational asymptotic method, using 3D elasticity theory as the
starting point. The self-contradictions of Kirchhoff assumptions and plane-stress assumptions used in both
Newtonian method and variational method are clearly pointed out. The
variational asymptotic method does not rely on any ad hoc assumptions, ending
in a self-consistent theory. It can be considered as a theoretical
tutorial for VAPAS, a general-purpose tool for modeling composite plates. 

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Jerry Brown's picture

Jerry Brown

Nice work. Are you aware of Novozhilov's work on plate deformation? In Foundations of the Nonlinear Theory of Elasticity, pgs 177-183 he presents an interesting method for deriving both the thin and thick plate models by setting the zz, xz and yz Green-Lagrange strains to zero and solving them to get estimates of z-axis displacement derivatives. 

Wenbin Yu's picture

Jerry, thanks a lot for your information. Yes, to avoid the contradiction of the ad hoc assumptions, one can abandon the starting kinematic assumptions and later back calculate the 3D strain field, and consequently the 3D displacement fields. My focus was trying to disclose the contradiction of what traditional textbooks have been teaching about as far as classical plate theory is concerned.

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