Mindlin–Reissner plate theory
Encyclopedia
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 1/10th 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 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 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.
where
and 
are the Cartesian coordinates on the mid-surface of the undeformed plate and 
is the coordinate for the thickness direction, 
are the in-plane displacements of the mid-surface,
is the displacement of the mid-surface in the 
direction, 
and 
designate the angles which the normal to the mid-surface makes with the 
axis. Unlike Kirchhoff-Love plate theory where 
are directly related to 
, Mindlin's theory requires that 
and 
.
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. 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 (
) is applied so that the correct amount of internal energy is predicted by the theory. Then
where
is an applied out-of-plane load, the in-plane stress resultants are defined as
the moment resultants are defined as
and the shear resultants are defined as
If the only external force is a vertical force on the top surface of the plate, the boundary conditions are
Since
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 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
in the plane of the plate are
where
is the Young's modulus, 
is the Poisson's ratio, and
are the in-plane strains. The through-the-thickness shear
stresses and strains are related by
where
is the shear modulus.
and
The bending rigidity is defined as the quantity
For a plate of thickness
, the bending rigidity has the form
In terms of the generalized deformations, these equations can be written as
The boundary conditions along the edges of a rectangular plate are
plates can be expressed as
Note that the plate thickness is
(and not 
) in the above equations and
. 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,
is the transverse displacement of the mid-surface of the plate
and the quantities
and 
are the rotations of the mid-surface normal
about the
and 
-axes, respectively. The canonical parameters for this theory
are
and 
. The shear correction factor 
usually has the
value
.
On the other hand, in Reissner's theory,
is the weighted average transverse deflection
while
and 
are equivalent rotations which are not identical to
those in Mindlin's theory. The canonical parameters for Reissner's theory are
, 
, and 
.
we can show that
where
is a biharmonic function such that 
. We can also
show that, if
is the displacement predicted for a Kirchhoff-Love plate,
where
is a function that satisfies the Laplace equation, 
. The
rotations of the normal are related to the displacements of a Kirchhoff-Love plate by
where
Kirchhoff–Love plate theory
The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love using assumptions...
that takes into account shear
Shear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
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 1/10th 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 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 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 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.Assumed displacement field
The Mindlin hypothesis implies that the displacements in the plate have the formwhere
and are the Cartesian coordinates on the mid-surface of the undeformed plate and is the coordinate for the thickness direction, are the in-plane displacements of the mid-surface, is the displacement of the mid-surface in the direction, and designate the angles which the normal to the mid-surface makes with the axis. Unlike Kirchhoff-Love plate theory where are directly related to , Mindlin's theory requires that and .
Strain-displacement 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 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. 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 (
) is applied so that the correct amount of internal energy is predicted by the theory. ThenEquilibrium equations
The equilibrium equations of a Mindlin-Reissner plate for small strains and small rotations have the formwhere
is an applied out-of-plane load, the in-plane stress resultants are defined asthe 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  and . 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 Calculus of variations Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown... , the equilibrium equations for a Mindlin-Reissner 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
Stress-strain relations
The stress-strain relations for a linear elastic Mindlin-Reissner plate are given bySince
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 stress-strain relations for an orthotropic materialOrthotropic material
An orthotropic material has two or three mutually orthogonal twofold axes of rotational symmetry so that its mechanical properties are, in general, different along each axis. Orthotropic materials are thus anisotropic; their properties depend on the direction in which they are measured...
, 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 stress-strain relationsin the plane of the plate are
where
is the Young's modulus, is the Poisson's ratio, and are the in-plane strains. The through-the-thickness shearstresses and strains are related by
where
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
, the bending rigidity has the formGoverning equations
If we ignore the in-plane extension of the plate, the governing equations areIn 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,  ,  , and  , we need athird 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 isotropicplates can be expressed as
Note that the plate thickness is
(and not ) in the above equations and. 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,
is the transverse displacement of the mid-surface of the plateand the quantities
and are the rotations of the mid-surface normalabout the
and -axes, respectively. The canonical parameters for this theoryare
and . The shear correction factor usually has thevalue
.On the other hand, in Reissner's theory,
is the weighted average transverse deflectionwhile
and are equivalent rotations which are not identical tothose in Mindlin's theory. The canonical parameters for Reissner's theory are
, , and .Relationship to Kirchhoff-Love theory
If we define the moment sum for Kirchhoff-Love theory aswe can show that
where
is a biharmonic function such that . We can alsoshow that, if
is the displacement predicted for a Kirchhoff-Love plate,where
is a function that satisfies the Laplace equation, . Therotations of the normal are related to the displacements of a Kirchhoff-Love plate by
where
See also
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- Infinitesimal strain theory
- Linear elasticityLinear elasticityLinear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
- Plate theoryPlate theoryIn continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions . The typical thickness to width ratio of a plate...
- Stress (mechanics)
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