Deformation of a plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue)
The Uflyand-Mindlin theory of vibrating 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 1948 by Yakov Solomonovich Uflyand (1916-1991) and in 1951 by Raymond Mindlin with Mindlin making reference to Uflyand's work. Hence, this theory has to be referred to us Uflyand-Mindlin plate theory, as is done in the handbook by Elishakoff, and in papers by Andronov, Elishakoff, Hache and Challamel, Loktev, Rossikhin and Shitikova and Wojnar. In 1994, Elishakoff suggested to neglect the fourth-order time derivative in Uflyand-Mindlin equations. A similar, but not identical, theory in static setting, 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 Uflyand-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 Uflyand-Mindlin plate theory that is most commonly used is actually due to Mindlin. The Reissner theory is slightly different and is a static counterpart of the Uflyand-Mindlin theory. Both theories include in-plane shear strains and both are extensions of Kirchhoff–Love plate theory incorporating first-order shear effects.
Uflyand-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 static 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 static theory does not invoke the plane stress condition.
The Uflyand-Mindlin 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 static plate theory.
Mindlin's theory was originally derived for isotropic plates using equilibrium considerations by Uflyand. 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 form
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 does not require that and .
Displacement of the mid-surface (left) and of a normal (right)
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. Then
The equilibrium equations of a Mindlin–Reissner plate for small strains and small rotations have the form
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
|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
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
Bending moments and normal stresses
Torques and shear stresses
Shear resultant and shear stresses
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. 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
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 relations
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.
The relations between the stress resultants and the generalized deformations are,
The bending rigidity is defined as the quantity
For a plate of thickness ( of the following all indicates thickness), the bending rigidity has the form
If we ignore the in-plane 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
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 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's static theory
The canonical constitutive relations for shear deformation theories of isotropic
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.
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
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.
Relationship to Kirchhoff–Love theory
If we define the moment sum for Kirchhoff–Love theory as
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
- ^ a b Uflyand, Ya. S.,1948, Wave Propagation by Transverse Vibrations of Beams and Plates, PMM: Journal of Applied Mathematics and Mechanics, Vol. 12, 287-300 (in Russian)
- ^ 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.
- ^ Elishakoff ,I.,2020, Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories, World Scientific, Singapore, ISBN 978-981-3236-51-6
- ^ Andronov, I.V.,2007, The Analytic Properties and Uniqueness of the Solutions To Problems of Scattering by Compact Obstacles in an Infinite Plate Described by the Uflyand-Mindlin Model, Acoustical Physics, Vol. 53(6), 653-659
- ^ Elishakoff, I., Hache, F. , Challamel N., 2017, Vibrations of Asymptotically and Variationally Based Uflyand-Mindlin Plate Models, International Journal of Engineering Science, Vol. 116, 58-73
- ^ Loktev, A.A.,2011, Dynamic Contact of a Spherical Center and Prestressed Orthtropic Uflyand-Mindlin Plate, Acta Mechanica, Vol. 222(1-2), 17-25
- ^ Rossikhin Y.A. and Shitikova M.V., Problem of the Impact Interaction of an Elastic Rod With a Uflyand-Mindlin Plate, International Applied Mechanics, Vol. 29(2), 118-125, 1993
- ^ Wojnar, R.,1979, Stress Equations of Motion for Uflyand-Mindlin Plate, Bulletin de l’ Academie Polonaise des Sciences – Serie des Sciences Techniques, Vol. 27(8-9), 731-740
- ^ Elishakoff, I, 1994, “Generalization of the Bolotin's dynamic edge effect method for vibration analysis of Mindlin plates,” Proceedings, the 1994 National Conference on Noise Control Engineering, (J. M. Cuschieri, S.A.L. Glegg and D.M. Yeager, eds.), New York, pp. 911 916
- ^ E. Reissner, 1945, The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics, Vol. 12, pp. A68–77.
- ^ 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,
- ^ These equations use a slightly different sign convention than
the preceding discussion.