 Mindlin–Reissner plate theory

The MindlinReissner theory of plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations throughthethickness of a plate. The theory was proposed in 1951 by Raymond Mindlin ^{[1]}. A similar, but not identical, theory had been proposed earlier by Eric Reissner in 1945 ^{[2]}. Both theories are intended for thick plates in which the normal to the midsurface remains straight but not necessarily perpendicular to the midsurface. The MindlinReissner theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of 1/10th the planar dimensions while the KirchhoffLove theory is applicable to thinner plates.
The form of MindlinReissner plate theory that is most commonly used is actually due to Mindlin and is more properly called Mindlin plate theory ^{[3]}. The Reissner theory is slightly different. Both theories include inplane shear strains and both are extensions of KirchhoffLove plate theory incorporating firstorder shear effects.
Mindlin's theory assumes that there is a linear variation of displacement across the plate thickness and but that the plate thickness does not change during deformation. This implies 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 throughthethickness 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 MindlinReissner theory is often called the firstorder shear deformation theory of plates. Since a firstorder shear deformation theory implies a linear displacement variation through the thickness, it is incompatible with Reissner's plate theory.
Contents
Mindlin 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]}.
Assumed displacement field
The Mindlin hypothesis implies that the displacements in the plate have the form
where x_{1} and x_{2} are the Cartesian coordinates on the midsurface of the undeformed plate and x_{3} is the coordinate for the thickness direction, are the inplane displacements of the midsurface, w^{0} is the displacement of the midsurface in the x_{3} direction, φ_{1} and φ_{2} designate the angles which the normal to the midsurface makes with the x_{3} axis. Unlike KirchhoffLove plate theory where φ_{α} are directly related to w^{0}, Mindlin's theory requires that and .
Straindisplacement relations
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 straindisplacement relations for MindlinReissner plates are
The shear strain, and hence the shear stress, across the thickness of the plate is not neglected in this theory. However, the shear strain is constant across the thickness of the plate. This cannot be accurate since the shear stress is known to be parabolic even for simple plate geometries. To account for the inaccuracy in the shear strain, a shear correction factor (κ) is applied so that the correct amount of internal energy is predicted by the theory. Then
Equilibrium equations
The equilibrium equations of a MindlinReissner plate for small strains and small rotations have the form
where q is an applied outofplane load, the inplane stress resultants are defined as
the moment resultants are defined as
and the shear resultants are defined as

Derivation of equilibrium equations For the situation where the strains and rotations of the plate are small the virtual internal energy is given by
where the stress resultants and stress moment resultants are defined in a way similar to that for Kirchhoff plates. The shear resultant is defined as
Integration by parts gives
The symmetry of the stress tensor implies that N_{αβ} = N_{βα} and M_{αβ} = M_{βα}. 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 MindlinReissner plate are
Boundary conditions
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
Stressstrain relations
The stressstrain relations for a linear elastic MindlinReissner plate are given by
Since σ_{33} does not appear in the equilibrium equations it is implicitly assumed that it do not have any effect on the momentum balance and is neglected. This assumption is also called the plane stress assumption. The remaining stressstrain 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
Mindlin theory for isotropic plates
For uniformly thick, homogeneous, and isotropic plates, the stressstrain relations in the plane of the plate are
where E is the Young's modulus, ν is the Poisson's ratio, and ε_{αβ} are the inplane strains. The throughthethickness shear stresses and strains are related by
where G = E / (2(1 + ν)) is the shear modulus.
Constitutive relations
The relations between the stress resultants and the generalized deformations are,
and
The bending rigidity is defined as the quantity
For a plate of thickness h, the bending rigidity has the form
Governing equations
If we ignore the inplane extension of the plate, the governing equations are
In terms of the generalized deformations, these equations can be written as

Derivation of equilibrium equations in terms of deformations If we expand out the governing equations of a Mindlin plate, we have
Recalling that
and combining the three governing equations, we have
If we define
we can write the above equation as
Similarly, using the relationships between the shear force resultants and the deformations, and the equation for the balance of shear force resultants, we can show that
Since there are three unknowns in the problem, φ_{1}, φ_{2}, and w^{0}, we need a third equation which can be found by differentiating the expressions for the shear force resultants and the governing equations in terms of the moment resultants, and equating these. 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
Relationship to Reissner theory
The canonical constitutive relations for shear deformation theories of isotropic plates can be expressed as^{[5]}^{[6]}
Note that the plate thickness is h (and not 2h) in the above equations and D = Eh^{3} / [12(1 − ν^{2})]. 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.
where
In Mindlin's theory, w^{0} is the transverse displacement of the midsurface of the plate and the quantities φ_{1} and φ_{2} are the rotations of the midsurface normal about the x_{2} and x_{1}axes, respectively. The canonical parameters for this theory are and . The shear correction factor κ usually has the value 5 / 6.
On the other hand, in Reissner's theory, w^{0} is the weighted average transverse deflection while φ_{1} and φ_{2} are equivalent rotations which are not identical to those in Mindlin's theory. The canonical parameters for Reissner's theory are , , and κ = 5 / 6.
Relationship to KirchhoffLove theory
If we define the moment sum for KirchhoffLove theory as
we can show that ^{[5]}
where Φ is a biharmonic function such that . We can also show that, if w^{K} is the displacement predicted for a KirchhoffLove plate,
where Ψ is a function that satisfies the Laplace equation, . The rotations of the normal are related to the displacements of a KirchhoffLove plate by
where
References
 ^ R. D. Mindlin, 1951, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, ASME Journal of Applied Mechanics, Vol. 18 pp. 31–38.
 ^ E. Reissner, 1945, The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics, Vol. 12, pp. A6877.
 ^ Wang, C. M., Lim, G. T., Reddy, J. N, Lee, K. H., 2001, Relationships between bending solutions of Reissner and Mindlin plate theories, Engineering Structures, vol. 23, pp. 838849.
 ^ Reddy, J. N., 1999, Theory and analysis of elastic plates, Taylor and Francis, Philadelphia.
 ^ ^{a} ^{b} Lim, G. T. and Reddy, J. N., 2003, On canonical bending relationships for plates, International Journal of Solids and Structures, vol. 40, pp. 30393067.
 ^ These equations use a slightly different sign convention than the preceding discussion.
See also
 Bending
 Bending of plates
 Infinitesimal strain theory
 Linear elasticity
 Plate theory
 Stress (mechanics)
 Vibration of plates
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