Mindlin–Reissner plate theory

The Mindlin–Reissner theory of plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1951 by Raymond Mindlin. A similar, but not identical, theory had been proposed earlier by Eric Reissner in 1945. Both theories are intended for thick plates in which the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. The Mindlin–Reissner theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of one tenth the planar dimensions while the Kirchhoff-Love theory is applicable to thinner plates.

The form of Mindlin–Reissner plate theory that is most commonly used is actually due to Mindlin and is more properly called Mindlin plate theory. The Reissner theory is slightly different. Both theories include in-plane shear strains and both are extensions of Kirchhoff-Love plate theory incorporating first-order shear effects.

Mindlin's theory assumes that there is a linear variation of displacement across the plate thickness but that the plate thickness does not change during deformation. An additional assumption is that the normal stress through the thickness is ignored; an assumption which is also called the plane stress condition. On the other hand, Reissner's theory assumes that the bending stress is linear while the shear stress is quadratic through the thickness of the plate. This leads to a situation where the displacement through-the-thickness is not necessarily linear and where the plate thickness may change during deformation. Therefore, Reissner's theory does not invoke the plane stress condition.

The Mindlin–Reissner theory is often called the first-order shear deformation theory of plates. Since a first-order shear deformation theory implies a linear displacement variation through the thickness, it is incompatible with Reissner's plate theory.

Mindlin's theory was originally derived for isotropic plates using equilibrium considerations. A more general version of the theory based on energy considerations is discussed here.

The Mindlin hypothesis implies that the displacements in the plate have the form

where ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ are the Cartesian coordinates on the mid-surface of the undeformed plate and ${\displaystyle x_{3}}$ is the coordinate for the thickness direction, ${\displaystyle u_{\alpha }^{0},~\alpha =1,2}$ are the in-plane displacements of the mid-surface, ${\displaystyle w^{0}}$ is the displacement of the mid-surface in the ${\displaystyle x_{3}}$ direction, ${\displaystyle \varphi _{1}}$ and ${\displaystyle \varphi _{2}}$ designate the angles which the normal to the mid-surface makes with the ${\displaystyle x_{3}}$ axis. Unlike Kirchhoff-Love plate theory where ${\displaystyle \varphi _{\alpha }}$ are directly related to ${\displaystyle w^{0}}$, Mindlin's theory requires that ${\displaystyle \varphi _{1}\neq w_{,1}^{0}}$ and ${\displaystyle \varphi _{2}\neq w_{,2}^{0}}$.

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