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[1] (1916-1991) and in 1951 by Raymond Mindlin[2] with Mindlin making reference to Uflyand's work. Hence, this theory has to be referred to as Uflyand-Mindlin plate theory, as is done in the handbook by Elishakoff,[3] and in papers by Andronov,[4] Elishakoff, Hache and Challamel,[5] Loktev,[6] Rossikhin and Shitikova[7] and Wojnar.[8] In 1994, Elishakoff[9] 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.[10] 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 theory[edit]
Mindlin's theory was originally derived for isotropic plates using equilibrium considerations by Uflyand.[1] A more general version of the theory based on energy considerations is discussed here.[11]
Assumed displacement field[edit]
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)
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Strain-displacement relations[edit]
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

Equilibrium equations[edit]
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
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For the situation where the strains and rotations of the plate are small the virtual internal energy is given by
![{\begin{aligned}\delta U&=\int _{{\Omega ^{0}}}\int _{{-h}}^{h}{\boldsymbol {\sigma }}:\delta {\boldsymbol {\epsilon }}~dx_{3}~d\Omega =\int _{{\Omega ^{0}}}\int _{{-h}}^{h}\left[\sigma _{{\alpha \beta }}~\delta \varepsilon _{{\alpha \beta }}+2~\sigma _{{\alpha 3}}~\delta \varepsilon _{{\alpha 3}}\right]~dx_{3}~d\Omega \\&=\int _{{\Omega ^{0}}}\int _{{-h}}^{h}\left[{\frac {1}{2}}~\sigma _{{\alpha \beta }}~(\delta u_{{\alpha ,\beta }}^{0}+\delta u_{{\beta ,\alpha }}^{0})-{\frac {x_{3}}{2}}~\sigma _{{\alpha \beta }}~(\delta \varphi _{{\alpha ,\beta }}+\delta \varphi _{{\beta ,\alpha }})+\kappa ~\sigma _{{\alpha 3}}\left(\delta w_{{,\alpha }}^{0}-\delta \varphi _{\alpha }\right)\right]~dx_{3}~d\Omega \\&=\int _{{\Omega ^{0}}}\left[{\frac {1}{2}}~N_{{\alpha \beta }}~(\delta u_{{\alpha ,\beta }}^{0}+\delta u_{{\beta ,\alpha }}^{0})-{\frac {1}{2}}M_{{\alpha \beta }}~(\delta \varphi _{{\alpha ,\beta }}+\delta \varphi _{{\beta ,\alpha }})+Q_{\alpha }\left(\delta w_{{,\alpha }}^{0}-\delta \varphi _{\alpha }\right)\right]~d\Omega \end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b294f8b86eafebb490f2157ab782eb1c75017c2)
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
![{\begin{aligned}\delta U&=\int _{{\Omega ^{0}}}\left[-{\frac {1}{2}}~(N_{{\alpha \beta ,\beta }}~\delta u_{{\alpha }}^{0}+N_{{\alpha \beta ,\alpha }}~\delta u_{{\beta }}^{0})+{\frac {1}{2}}(M_{{\alpha \beta ,\beta }}~\delta \varphi _{{\alpha }}+M_{{\alpha \beta ,\alpha }}\delta \varphi _{{\beta }})-Q_{{\alpha ,\alpha }}~\delta w^{0}-Q_{\alpha }~\delta \varphi _{\alpha }\right]~d\Omega \\&+\int _{{\Gamma ^{0}}}\left[{\frac {1}{2}}~(n_{\beta }~N_{{\alpha \beta }}~\delta u_{\alpha }^{0}+n_{\alpha }~N_{{\alpha \beta }}~\delta u_{{\beta }}^{0})-{\frac {1}{2}}(n_{\beta }~M_{{\alpha \beta }}~\delta \varphi _{{\alpha }}+n_{\alpha }M_{{\alpha \beta }}\delta \varphi _{\beta })+n_{\alpha }~Q_{\alpha }~\delta w^{0}\right]~d\Gamma \end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/babf6574b41c661c704d8c39ea86e4d4cbef63b9)
The symmetry of the stress tensor implies that and
. Hence,
![{\begin{aligned}\delta U&=\int _{{\Omega ^{0}}}\left[-N_{{\alpha \beta ,\alpha }}~\delta u_{{\beta }}^{0}+\left(M_{{\alpha \beta ,\beta }}-Q_{\alpha }\right)~\delta \varphi _{{\alpha }}-Q_{{\alpha ,\alpha }}~\delta w^{0}\right]~d\Omega \\&+\int _{{\Gamma ^{0}}}\left[n_{\alpha }~N_{{\alpha \beta }}~\delta u_{{\beta }}^{0}-n_{\beta }~M_{{\alpha \beta }}~\delta \varphi _{{\alpha }}+n_{\alpha }~Q_{\alpha }~\delta w^{0}\right]~d\Gamma \end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79d08f092a2df2f126e0441e8d3974c08ea6f4e)
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,
![{\begin{aligned}&\int _{{\Omega ^{0}}}\left[N_{{\alpha \beta ,\alpha }}~\delta u_{{\beta }}^{0}-\left(M_{{\alpha \beta ,\beta }}-Q_{\alpha }\right)~\delta \varphi _{{\alpha }}+\left(Q_{{\alpha ,\alpha }}+q\right)~\delta w^{0}\right]~d\Omega \\&\qquad \qquad =\int _{{\Gamma ^{0}}}\left[n_{\alpha }~N_{{\alpha \beta }}~\delta u_{{\beta }}^{0}-n_{\beta }~M_{{\alpha \beta }}~\delta \varphi _{{\alpha }}+n_{\alpha }~Q_{\alpha }~\delta w^{0}\right]~d\Gamma \end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16de140c7eb087e78e035e0ef025f2bd97ef0310)
Using standard arguments from the calculus of variations, the equilibrium equations for a Mindlin–Reissner plate are

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Bending moments and normal stresses
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Torques and shear stresses
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Shear resultant and shear stresses
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Boundary conditions[edit]
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[edit]
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

Then
![{\displaystyle {\begin{aligned}{\begin{bmatrix}N_{11}\\N_{22}\\N_{12}\end{bmatrix}}&=\int _{-h}^{h}{\begin{bmatrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{12}\end{bmatrix}}dx_{3}\\[5pt]&=\left\{\int _{-h}^{h}{\begin{bmatrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\0&0&C_{66}\end{bmatrix}}~dx_{3}\right\}{\begin{bmatrix}u_{1,1}^{0}\\u_{2,2}^{0}\\{\frac {1}{2}}~(u_{1,2}^{0}+u_{2,1}^{0})\end{bmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2773bd799eaf540f6d2a0b63459949c35498c40)
and
![{\displaystyle {\begin{aligned}{\begin{bmatrix}M_{11}\\M_{22}\\M_{12}\end{bmatrix}}&=\int _{-h}^{h}x_{3}~{\begin{bmatrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\0&0&C_{66}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{12}\end{bmatrix}}dx_{3}\\[5pt]&=-\left\{\int _{-h}^{h}x_{3}^{2}~{\begin{bmatrix}C_{11}&C_{12}&0\\C_{12}&C_{22}&0\\0&0&C_{66}\end{bmatrix}}~dx_{3}\right\}{\begin{bmatrix}\varphi _{1,1}\\\varphi _{2,2}\\{\frac {1}{2}}(\varphi _{1,2}+\varphi _{2,1})\end{bmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35f69973cdf7bd4fb0da3542e1dd3fc08b4ddb79)
For the shear terms

The extensional stiffnesses are the quantities

The bending stiffnesses are the quantities

Mindlin theory for isotropic plates[edit]
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.
Constitutive relations[edit]
The relations between the stress resultants and the generalized deformations are,
![{\displaystyle {\begin{aligned}{\begin{bmatrix}N_{11}\\N_{22}\\N_{12}\end{bmatrix}}&={\cfrac {2Eh}{1-\nu ^{2}}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&1-\nu \end{bmatrix}}{\begin{bmatrix}u_{1,1}^{0}\\u_{2,2}^{0}\\{\frac {1}{2}}~(u_{1,2}^{0}+u_{2,1}^{0})\end{bmatrix}},\\[5pt]{\begin{bmatrix}M_{11}\\M_{22}\\M_{12}\end{bmatrix}}&=-{\cfrac {2Eh^{3}}{3(1-\nu ^{2})}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&1-\nu \end{bmatrix}}{\begin{bmatrix}\varphi _{1,1}\\\varphi _{2,2}\\{\frac {1}{2}}(\varphi _{1,2}+\varphi _{2,1})\end{bmatrix}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d361c2d0da765cf68fb6344e6d8f1af5189afbe8)
and

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

Governing equations[edit]
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
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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 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

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The boundary conditions along the edges of a rectangular plate are

Relationship to Reissner's static theory[edit]
The canonical constitutive relations for shear deformation theories of isotropic
plates can be expressed as[12][13]
![{\displaystyle {\begin{aligned}M_{11}&=D\left[{\mathcal {A}}\left({\frac {\partial \varphi _{1}}{\partial x_{1}}}+\nu {\frac {\partial \varphi _{2}}{\partial x_{2}}}\right)-(1-{\mathcal {A}})\left({\frac {\partial ^{2}w^{0}}{\partial x_{1}^{2}}}+\nu {\frac {\partial ^{2}w^{0}}{\partial x_{2}^{2}}}\right)\right]+{\frac {q}{1-\nu }}\,{\mathcal {B}}\\[5pt]M_{22}&=D\left[{\mathcal {A}}\left({\frac {\partial \varphi _{2}}{\partial x_{2}}}+\nu {\frac {\partial \varphi _{1}}{\partial x_{1}}}\right)-(1-{\mathcal {A}})\left({\frac {\partial ^{2}w^{0}}{\partial x_{2}^{2}}}+\nu {\frac {\partial ^{2}w^{0}}{\partial x_{1}^{2}}}\right)\right]+{\frac {q}{1-\nu }}\,{\mathcal {B}}\\[5pt]M_{12}&={\frac {D(1-\nu )}{2}}\left[{\mathcal {A}}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}+{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)-2(1-{\mathcal {A}})\,{\frac {\partial ^{2}w^{0}}{\partial x_{1}\partial x_{2}}}\right]\\Q_{1}&={\mathcal {A}}\kappa Gh\left(\varphi _{1}+{\frac {\partial w^{0}}{\partial x_{1}}}\right)\\[5pt]Q_{2}&={\mathcal {A}}\kappa Gh\left(\varphi _{2}+{\frac {\partial w^{0}}{\partial x_{2}}}\right)\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9359b1339f684729b029ff67700f1f0f9509d7a)
Note that the plate thickness is
(and not
) in the above equations and
. If we define a Marcus moment,
![{\mathcal {M}}=D\left[{\mathcal {A}}\left({\frac {\partial \varphi _{1}}{\partial x_{1}}}+{\frac {\partial \varphi _{2}}{\partial x_{2}}}\right)-(1-{\mathcal {A}})\nabla ^{2}w^{0}\right]+{\frac {2q}{1-\nu ^{2}}}{\mathcal {B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3865d48fc39866c6ad36278341d2888728d30e9)
we can express the shear resultants as
![{\displaystyle {\begin{aligned}Q_{1}&={\frac {\partial {\mathcal {M}}}{\partial x_{1}}}+{\frac {D(1-\nu )}{2}}\left[{\mathcal {A}}{\frac {\partial }{\partial x_{2}}}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}-{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)\right]-{\frac {\mathcal {B}}{1+\nu }}{\frac {\partial q}{\partial x_{1}}}\\[5pt]Q_{2}&={\frac {\partial {\mathcal {M}}}{\partial x_{2}}}-{\frac {D(1-\nu )}{2}}\left[{\mathcal {A}}{\frac {\partial }{\partial x_{1}}}\left({\frac {\partial \varphi _{1}}{\partial x_{2}}}-{\frac {\partial \varphi _{2}}{\partial x_{1}}}\right)\right]-{\frac {\mathcal {B}}{1+\nu }}{\frac {\partial q}{\partial x_{2}}}\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17cba4884439133de00955b12c38f2795efeed27)
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.
Relationship to Kirchhoff–Love theory[edit]
If we define the moment sum for Kirchhoff–Love theory as

we can show that [12]

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

References[edit]
- ^ 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,
pp. 3039–3067.
- ^ These equations use a slightly different sign convention than
the preceding discussion.
See also[edit]