Let us consider bending of plates. Transverse stresses shall vanish (or at least be negligibly small) in comparison to the in-plane ones at bending of thin structures, see "Asymptotic splitting in the three-dimensional problem of elasticity for non-homogeneous piezoelectric plates" by Yury Vetyukov, Alexey Kuzin and Michael Krommer. This condition needs to be fulfilled by the interpolation. Thus, if the in-plane displacements change linear over the thickness, then the in-plane strains are linear in this direction. This means that the transverse strains need also be linear, and the transverse displacement needs to be quadratic for obtaining an accurate solution. A separate approximation needs to be chosen for each layer owing to different material properties in the relation between the in-plane and transverse strains.

This complexity may be avoided by using so-called "assumed strain" approach, in which one does not approximate the variation of the transverse displacement over the thickness, but uses one constant value in some reference layer for describing the kinematics of the deformation and then uses constitutive relations such as if the above mentioned condition was fulfilled.

Thank you so much

Interesting.