In case of plane stress and plane strain , the biharmonic equation roots will be different as the Hooke's law is different for both the cases. So, I would like to ask can we use same complex potential function (identical in form) for both the cases , which eventually will be different and produce distinct stress / displacement field .
As the in-plane stress distribution in both cases is not depend on the Hooke's law ( this follows from the equilibrium equations), we can use the same complex potential function.
Derive a suitable complex potential function for each case of plane stress and plane strain based on the foundational equations of 2d elasticity theory, and the answer becomes obvious.
Yes. The same Complex Potential Function can be used for both plane stress and plane strain.
This is because in the absence of body forces the Airy Stress functions for plane stress and plane strain are governed by the same biharmonic equation
∇4Ф = 0.
The only difference is in the expression for displacements (strains). Those for plane stress can be derived from those for plane strain by replacing Poisson's ratio υ by υ* where υ* = υ /(1+ υ)
One further comment should be made. Generally speaking, the stresses distribution may depend on elastic constants for multiply connected 2-d regions (see Theory of elasticity / S.P. Timoshenko, J. N. Goodier.)
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