Mindlin–Reissner Plate Theory

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 UflyandUflyand, 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) (1916-1991) and in 1951 by Raymond Mindlin 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, 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 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
:$\begin u_\alpha(\mathbf) & = u^0_\alpha(x_1,x_2) - x_3~\varphi_\alpha ~;~~\alpha=1,2 \\ u_3(\mathbf) & = w^0(x_1, x_2) \end$
where $x_1$ and $x_2$ are the Cartesian coordinates on the mid-surface of the undeformed plate and $x_3$ is the coordinate for the thickness direction, $u^0_\alpha,~ \alpha=1,2$ are the in-plane displacements of the mid-surface,
$w^0$ is the displacement of the mid-surface in the $x_3$ direction, $\varphi_1$ and $\varphi_2$ designate the angles which the normal to the mid-surface makes with the $x_3$ axis. Unlike Kirchhoff–Love plate theory where $\varphi_\alpha$ are directly related to $w^0$, Mindlin's theory does not require that $\varphi_1 = w^0_$ and $\varphi_2 = w^0_$.

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
:$\begin \varepsilon_ & = \frac(u^0_+u^0_) - \frac~(\varphi_+\varphi_) \\ \varepsilon_ & = \cfrac\left(w^0_- \varphi_\alpha\right) \\ \varepsilon_ & = 0 \end$
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 ($\kappa$) is applied so that the correct amount of internal energy is predicted by the theory. Then
:$\varepsilon_ = \cfrac~\kappa~\left(w^0_- \varphi_\alpha\right)$

Equilibrium equations

The equilibrium equations of a Mindlin–Reissner plate for small strains and small rotations have the form
:$\begin & N_ = 0 \\ & M_-Q_\alpha = 0 \\ & Q_+q = 0 \end$
where $q$ is an applied out-of-plane load, the in-plane stress resultants are defined as
:$N_ := \int_^h \sigma_~dx_3 \,,$
the moment resultants are defined as
:$M_ := \int_^h x_3~\sigma_~dx_3 \,,$
and the shear resultants are defined as
:$Q_\alpha := \kappa~\int_^h \sigma_~dx_3 \,.$
:

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
:$\begin n_\alpha~N_ & \quad \mathrm \quad u^0_\beta \\ n_\alpha~M_ & \quad \mathrm \quad \varphi_\alpha \\ n_\alpha~Q_\alpha & \quad \mathrm \quad w^0 \end$

Stress–strain relations

The stress–strain relations for a linear elastic Mindlin–Reissner plate are given by
:$\begin \sigma_ & = C_~\varepsilon_ \\ \sigma_ & = C_~\varepsilon_ \\ \sigma_ & = C_~\varepsilon_ \end$
Since $\sigma_$ 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
:$\begin\sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \\ \sigma_ \end = \begin C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & 0 & 0 & 0 \\ 0 & 0 & C_ & 0 & 0 \\ 0 & 0 & 0 & C_ & 0 \\ 0 & 0 & 0 & 0 & C_\end \begin\varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \\\varepsilon_\end$
Then
:
\begin
\beginN_ \\ N_ \\ N_ \end & =
\int_^h \begin C_ & C_ & 0 \\ C_ & C_ & 0 \\
0 & 0 & C_ \end
\begin\varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \end
dx_3 \\ pt& = \left\
\begin u^0_ \\ u^0_ \\ \frac~(u^0_+u^0_) \end
\end

and
:
\begin
\beginM_ \\ M_ \\ M_ \end & =
\int_^h x_3~\begin C_ & C_ & 0 \\ C_ & C_ & 0 \\
0 & 0 & C_ \end
\begin\varepsilon_ \\ \varepsilon_ \\ \varepsilon_ \end
dx_3 \\ pt& = -\left\
\begin \varphi_ \\ \varphi_ \\ \frac(\varphi_+\varphi_) \end
\end

For the shear terms
:$\beginQ_1 \\ Q_2 \end = \kappa~\int_^h \begin C_ & 0 \\ 0 & C_ \end \begin\varepsilon_ \\ \varepsilon_ \end dx_3 = \cfrac\left\ \begin w^0_ - \varphi_1 \\ w^0_ - \varphi_2 \end$
The extensional stiffnesses are the quantities
:$A_ := \int_^h C_~dx_3$
The bending stiffnesses are the quantities
:$D_ := \int_^h x_3^2~C_~dx_3 \,.$

Mindlin theory for isotropic plates

For uniformly thick, homogeneous, and isotropic plates, the stress–strain relations
in the plane of the plate are
:$\begin\sigma_ \\ \sigma_ \\ \sigma_ \end = \cfrac \begin 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \cfrac \end \begin\varepsilon_ \\ \varepsilon_ \\ 2\varepsilon_ \end \,.$
where $E$ is the Young's modulus, $\nu$ is the Poisson's ratio, and
$\varepsilon_$ are the in-plane strains. The through-the-thickness shear
stresses and strains are related by
:$\sigma_ = 2G\varepsilon_ \quad \text \quad \sigma_ = 2G\varepsilon_$
where $G = E/(2(1+\nu))$ is the shear modulus.

Constitutive relations

The relations between the stress resultants and the generalized deformations are,
:
\begin
\beginN_ \\ N_ \\ N_ \end & =
\cfrac \begin 1 & \nu & 0 \\ \nu & 1 & 0 \\
0 & 0 & 1-\nu \end
\begin u^0_ \\ u^0_ \\ \frac~(u^0_+u^0_) \end, \\ pt \beginM_ \\ M_ \\ M_ \end & =
-\cfrac \begin 1 & \nu & 0 \\ \nu & 1 & 0 \\
0 & 0 & 1-\nu \end
\begin \varphi_ \\ \varphi_ \\ \frac(\varphi_+\varphi_) \end,
\end

and
:$\beginQ_1 \\ Q_2 \end = \kappa G 2 h \begin w^0_ - \varphi_1 \\ w^0_ - \varphi_2 \end \,.$
The bending rigidity is defined as the quantity
:$D = \cfrac \,.$
For a plate of thickness $h$ ($h$ of the following all indicates thickness), the bending rigidity has the form
:$D = \cfrac \,.$

Governing equations

If we ignore the in-plane extension of the plate, the governing equations are
:$\begin M_-Q_\alpha & = 0 \\ Q_+q & = 0 \,. \end$
In terms of the generalized deformations, these equations can be written as
:$\begin &\nabla^2 \left(\frac + \frac\right) = \frac \\ &\nabla^2 w^0 - \frac - \frac = -\frac \\ &\nabla^2 \left(\frac - \frac\right) = \frac\left(\frac - \frac\right) \,. \end$
:
The boundary conditions along the edges of a rectangular plate are
:$\begin \text \quad & \quad w^0 = 0, M_ = 0 ~(\text~M_ = 0), \varphi_1 = 0 ~(\text \varphi_2 = 0) \\ \text \quad & \quad w^0 = 0, \varphi_1 = 0, \varphi_ = 0 \,. \end$

Relationship to Reissner's static theory

The canonical constitutive relations for shear deformation theories of isotropic
plates can be expressed asLim, 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.
:
\begin
M_ & = D\left mathcal\left(\frac+\nu\frac\right)
- (1-\mathcal)\left(\frac + \nu\frac\right)\right
+ \frac\,\mathcal\\ pt M_ & = D\left mathcal\left(\frac+\nu\frac\right)
- (1-\mathcal)\left(\frac + \nu\frac\right)\right
+ \frac\,\mathcal\\ pt M_ & = \frac\left mathcal\left(\frac+\frac\right)
- 2(1-\mathcal)\,\frac\right \\
Q_1 & = \mathcal \kappa G h\left(\varphi_1 + \frac\right) \\ pt Q_2 & = \mathcal \kappa G h\left(\varphi_2 + \frac\right) \,.
\end

Note that the plate thickness is $h$ (and not $2h$) in the above equations and
D = Eh^3/ 2(1-\nu^2)/math>. If we define a ''Marcus moment'',
:
\mathcal = D\left mathcal\left(\frac + \frac\right)
- (1-\mathcal)\nabla^2 w^0\right+ \frac\mathcal

we can express the shear resultants as
:
\begin
Q_1 & = \frac
+ \frac\left mathcal\frac\left(\frac
-\frac\right)\right- \frac\frac \\ pt Q_2 & = \frac
- \frac\left mathcal\frac\left(\frac
-\frac\right)\right- \frac\frac\,.
\end

These relations and the governing equations of equilibrium, when combined, lead to the
following canonical equilibrium equations in terms of the generalized displacements.
:$\begin & \nabla^2 \left(\mathcal - \frac\,q\right) = -q \\ & \kappa G h\left(\nabla^2 w^0 + \frac\right) = -\left(1 - \cfrac\right)q \\ & \nabla^2 \left(\frac - \frac\right) = c^2\left(\frac - \frac\right) \end$
where
:$c^2 = \frac \,.$
In Mindlin's theory, $w^0$ is the transverse displacement of the mid-surface of the plate
and the quantities $\varphi_1$ and $\varphi_2$ are the rotations of the mid-surface normal
about the $x_2$ and $x_1$-axes, respectively. The canonical parameters for this theory
are $\mathcal = 1$ and $\mathcal = 0$. The shear correction factor $\kappa$ usually has the
value $5/6$.

On the other hand, in Reissner's theory, $w^0$ is the weighted average transverse deflection
while $\varphi_1$ and $\varphi_2$ 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
:$\mathcal^K := -D\nabla^2 w^K$
we can show that
:$\mathcal = \mathcal^K + \frac\,q + D \nabla^2 \Phi$
where $\Phi$ is a biharmonic function such that $\nabla^2 \nabla^2 \Phi = 0$. We can also
show that, if $w^K$ is the displacement predicted for a Kirchhoff–Love plate,
:$w^0 = w^K + \frac\left(1 - \frac\right) - \Phi + \Psi$
where $\Psi$ is a function that satisfies the Laplace equation, $\nabla^2 \Psi = 0$. The
rotations of the normal are related to the displacements of a Kirchhoff–Love plate by
:$\begin \varphi_1 = - \frac - \frac\left(1 - \frac - \frac\right)Q_1^K + \frac\left(\frac\nabla^2 \Phi + \Phi - \Psi\right) + \frac\frac \\ \varphi_2 = - \frac - \frac\left(1 - \frac - \frac\right)Q_2^K + \frac\left(\frac\nabla^2 \Phi + \Phi - \Psi\right) + \frac\frac \end$
where
:$Q_1^K = -D\frac\left(\nabla^2 w^K\right) ~,~~ Q_2^K = -D\frac\left(\nabla^2 w^K\right) ~,~~ \Omega := \frac - \frac \,.$

References

See also

*Bending
*Bending of plates
*Infinitesimal strain theory
*Linear elasticity
*Plate theory
*Stress (mechanics)
*Stress resultants
*Vibration of plates

{{DEFAULTSORT:Mindlin-Reissner plate theory
Continuum mechanics