Reissner-Mindlin plate theory

[1] A similar, but not identical, theory in static setting, had been proposed earlier by Eric Reissner in 1945.

The Reissner-Mindlin 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 Reissner-Mindlin plate theory that is most commonly used is actually due to Mindlin and is more properly called Mindlin plate theory.

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.

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 static theory does not invoke the plane stress condition.

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.

[4] The Mindlin hypothesis implies that the displacements in the plate have the form where

are the Cartesian coordinates on the mid-surface of the undeformed plate and

designate the angles which the normal to the mid-surface makes with the

Depending on the amount of rotation of the plate normals two different approximations for the strains can be derived from the basic kinematic assumptions.

For small strains and small rotations the strain–displacement relations for Mindlin–Reissner plates are The shear strain, and hence the shear stress, across the thickness of the plate is not neglected in this theory.

This cannot be accurate since the shear stress is known to be parabolic even for simple plate geometries.

) is applied so that the correct amount of internal energy is predicted by the theory.

The shear resultant is defined as Integration by parts gives The symmetry of the stress tensor implies that

Hence, For the special case when the top surface of the plate is loaded by a force per unit area

, the virtual work done by the external forces is Then, from the principle of virtual work, Using standard arguments from the calculus of variations, the equilibrium equations for a Mindlin–Reissner plate are The boundary conditions are indicated by the boundary terms in the principle of virtual work.

If the only external force is a vertical force on the top surface of the plate, the boundary conditions are The stress–strain relations for a linear elastic Mindlin–Reissner plate are given by Since

does not appear in the equilibrium equations it is implicitly assumed that it does not have any effect on the momentum balance and is neglected.

The remaining stress–strain relations for an orthotropic material, in matrix form, can be written as Then and For the shear terms The extensional stiffnesses are the quantities The bending stiffnesses are the quantities For uniformly thick, homogeneous, and isotropic plates, the stress–strain relations in the plane of the plate are where

The through-the-thickness shear stresses and strains are related by where

, the bending rigidity has the form from now on, in all the equations below, we will refer to

as the total thickness of the plate, and as not the semi-thickness (as in the above equations).

The resulting equation has the form Therefore, the three governing equations in terms of the deformations are The boundary conditions along the edges of a rectangular plate are The canonical constitutive relations for shear deformation theories of isotropic plates can be expressed as[5][6] Note that the plate thickness is

If we define a Marcus moment, we can express the shear resultants as These relations and the governing equations of equilibrium, when combined, lead to the following canonical equilibrium equations in terms of the generalized displacements.

are equivalent rotations which are not identical to those in Mindlin's theory.

If we define the moment sum for Kirchhoff–Love theory as we can show that [5] where