Chapter 10

Approximate theories for solids with special shapes:

rods, beams, membranes, plates and shells

10.6 Simplified versions of general shell theory $–$ flat plates and membranes

In many practical cases of interest the general equations of shell theory can be vastly simplified.  In this section, we summarize the governing equations for a number of special solids, including equations governing behavior of flat plates, and membranes.

This is the simplest version of plate theory, and is used in most practical applications. The figure illustrates the problem to be solved.  An initially flat plate, which has uniform (small) thickness h, Young’s modulus $E$ Poisson’s ratio $\nu$ and mass density $\rho$,  is subjected to a distributed force $p$ per unit area (acting vertically upwards).  The edge of the plate can be constrained in various ways, as discussed in more detail below.   We wish to determine its deformed shape, as well as the internal forces and moments in the plate.

All vector and tensor quantities can be expressed in a fixed Cartesian basis $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ illustrated in the figure.  The covariant and reciprocal basis vectors are identical so there is no need to use the system of raised and lowered indices that was needed in general shell theory. In addition, since the basis vectors are independent of position, the Christoffel symbols are all zero.  We continue to use the convention that Greek subscripts can have values 1 or 2, while Latin subscripts may have values 1,2,3.

We make the following assumptions

1. The variation of displacements within the plate conforms to the usual approximations of Kirchhoff plate theory, i.e. material fibers that are perpendicular to the mid-plane of the undeformed plate remain perpendicular to the mid-plane of the deformed plate, and stretching transverse to the mid-plane is neglected.
2. The displacement of the mid-plane of the plate has the form $u={u}_{3}\left({x}_{1},{x}_{2}\right){e}_{3}$, i.e. material points on the mid-plane of the plate deflect only transverse to the plate.
3. The mid-plane deflection is small compared with the dimensions of the plate, and the slope of the deflected plate is small, so that $\partial {u}_{3}/\partial {x}_{\alpha }<<1$ for all $\alpha$; second order terms in displacement are ignored in all the strain definitions and also the equilibrium equations.

The following (approximate) results can then be extracted from the general shell equations:

Kinematics:

The curvature change tensor is $\Delta {\kappa }_{\alpha \beta }=-\frac{{\partial }^{2}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\beta }}$, while the in-plane strain tensor is ${\gamma }_{\alpha \beta }=0$.

The normal vector to the deformed plate can be approximated as ${m}_{3}={e}_{3}-\frac{\partial {u}_{3}}{\partial {\xi }_{\alpha }}{e}_{\alpha }$

The displacement field in the plate is approximated as $u={u}_{3}{e}_{3}-{x}_{3}\frac{\partial {u}_{3}}{\partial {x}_{\alpha }}{e}_{\alpha }$

The (infinitesimal) strain field in the plate is approximated as ${\epsilon }_{\alpha \beta }={x}_{3}\Delta {\kappa }_{\alpha \beta }$

Kinetics

The external force is characterized by the force per unit area $p{e}_{3}$ acting on the surface of the plate;

The in-plane stress tensor ${T}_{\alpha \beta }=0$, so the internal forces are completely characterized by the internal moment tensor $M={M}_{\alpha \beta }{e}_{\alpha }\otimes \left({m}_{3}×{e}_{\beta }\right)$ and transverse force tensor $V={V}_{\alpha }{e}_{\alpha }$;

The components ${V}_{\alpha }$ represent the vertical force per unit length acting on an internal plane perpendicular to the ${e}_{\alpha }$ direction;

The physical significance of ${M}_{\alpha \beta }$ is illustrated in the figure: ${M}_{1\alpha }$ characterizes the moment per unit length acting on planes inside the shell that are normal to the ${e}_{1}$ direction, while ${M}_{2\alpha }$ characterizes the moment per unit length acting on planes that are normal to ${e}_{2}$.  Note that ${M}_{\alpha 1}$ represents a moment about the ${e}_{2}$ axis, while ${M}_{\alpha 2}$ is a moment acting about the $-{e}_{1}$ axis.

Moment-Curvature relation reduces to ${M}_{\alpha \beta }=\frac{E{h}^{3}}{12\left(1-{\nu }^{2}\right)}\left(\left(1-\nu \right)\Delta {\kappa }_{\alpha \beta }+\nu \Delta {\kappa }_{\lambda \lambda }{\delta }_{\alpha \beta }\right)$

Equations of Motion  are approximated by

$\frac{\partial {V}_{\alpha }}{\partial {x}_{\alpha }}+p=\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}$  $\frac{\partial {M}_{\alpha \beta }}{\partial {x}_{\alpha }}-{V}_{\beta }\approx 0$

(rotational inertia has been neglected). These equations can be combined to eliminate V

$\frac{{\partial }^{2}{M}_{\alpha \beta }}{\partial {x}_{\alpha }\partial {x}_{\beta }}+p=\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\equiv \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}{M}_{11}}{\partial {x}_{1}^{2}}+2\frac{{\partial }^{2}{M}_{12}}{\partial {x}_{1}\partial {x}_{2}}+\frac{{\partial }^{2}{M}_{22}}{\partial {x}_{2}^{2}}+p=\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}$

and can also be expressed in terms of the displacements as

$\frac{E{h}^{3}}{12\left(1-{\nu }^{2}\right)}\frac{{\partial }^{4}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\alpha }\partial {x}_{\beta }\partial {x}_{\beta }}+\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}=\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\equiv \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{E{h}^{3}}{12\left(1-{\nu }^{2}\right)}\left\{\frac{{\partial }^{4}{u}_{3}}{\partial {x}_{1}^{4}}+2\frac{{\partial }^{4}{u}_{3}}{\partial {x}_{1}^{2}\partial {x}_{2}^{2}}+\frac{{\partial }^{4}{u}_{3}}{\partial {x}_{2}^{4}}\right\}+\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}=p$

Edge boundary conditions.  The edge of the plate is characterized by a curve C that lies in the mid-plane of the shell, encircling ${e}_{3}$ in a counterclockwise sense.   We let $s$ denote arc-length measured around C from some convenient origin, and use $\tau ={\tau }_{\alpha }{e}_{\alpha }$ and $n={e}_{3}×\tau$ denote unit vectors tangent and normal to C.  Elementary plate theory offers the following choices of boundary condition for each point on C:

1. Part of the boundary of the plate ${C}_{1}$ may be clamped, i.e. rotations and displacement of the boundary are completely prevented. The transverse displacement must then satisfy ${u}_{3}={n}_{\alpha }\partial {u}_{3}/d{x}_{\alpha }=0$ on ${C}_{1}$.
2. Part of the boundary ${C}_{2}$ may be simply supported, i.e. the boundary of the plate is prevented from moving, but is permitted to rotate freely about the tangent vector $\tau$.  In this case the transverse displacement and internal moment must satisfy

${u}_{3}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{n}_{\alpha }{M}_{\alpha \beta }{n}_{\beta }=0$

1. Part of the boundary ${C}_{3}$ may be free, i.e. the boundary is free to both translate and rotate.  In this case the transverse shear force and internal moment must satisfy

${n}_{\alpha }{V}_{\alpha }+\frac{\partial }{\partial s}\left[{n}_{\alpha }{M}_{\alpha \beta }{\tau }_{\beta }\right]=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{n}_{\alpha }{M}_{\alpha \beta }{n}_{\beta }=0$

More general boundary conditions, in which the edge of the plate is subjected to prescribed forces and moments, can also be derived from the results given in Section 10.5.8 if this is of interest.

Strain energy and kinetic energy of a flat plate: The formula for the strain energy and kinetic energy of the plate can be expressed in terms of displacements as

$\begin{array}{l}\Phi =\frac{E{h}^{3}}{24\left(1-{\nu }^{2}\right)}\underset{A}{\int }\left(\left(1-\nu \right)\frac{{\partial }^{2}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\beta }}\frac{{\partial }^{2}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\beta }}+\nu {\left(\frac{{\partial }^{2}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\alpha }}\right)}^{2}\right)dA\\ K=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{A}{\int }\left(\frac{h}{2}\rho {v}_{i}{v}_{i}+\frac{{h}^{3}}{24}\rho \frac{\partial {v}_{3}}{\partial {x}_{\alpha }}\frac{\partial {v}_{3}}{\partial {x}_{\alpha }}\right)dA\end{array}$

The second term in the integral for the kinetic energy represents the energy associated with the plate’s out-of-plane rotation, and can be ignored in most practical applications.

This version of plate theory is used to model plates that are subjected to substantial loading parallel to the plane of the plate (usually due to loads applied at its boundaries).  The theory assumes that displacements are small enough to use linearized measures of strain, but includes nonlinear terms associated with the in-plane loading in the equilibrium equations.  The theory can be used to calculate buckling loads for plates, but does not accurately model their deformation if the buckling loads are exceeded.

The problem to be solved is stated in Sect 10.6.1.   The majority of the governing equations are identical to those of standard plate theory, given in 10.6.1 $–$ the equations which need to be modified to account for transverse forces are listed below.

Kinematics:

The in-plane strain tensor is approximated as ${\gamma }_{\alpha \beta }=\left(\partial {u}_{\alpha }/\partial {u}_{\beta }+\partial {u}_{\beta }/\partial {u}_{\alpha }\right)/2$

The displacement field in the plate is approximated as $u={u}_{i}{e}_{i}-{x}_{3}\frac{\partial {u}_{3}}{\partial {x}_{\alpha }}{e}_{\alpha }$

The (infinitesimal) strain field in the plate is approximated as ${\epsilon }_{\alpha \beta }={\gamma }_{\alpha \beta }+{x}_{3}\Delta {\kappa }_{\alpha \beta }$, where the components of the curvature change tensor are given in 10.6.1.

Kinetics

The external force consists of a force per unit area ${p}_{i}{e}_{i}$ acting on the surface of the plate.

The internal forces are characterized as described in 10.6.1, except that the stress resultant tensor ${T}_{\alpha \beta }$ is nonzero. The components of the stress resultant tensor can be interpreted as illustrated in the figure: ${T}_{\alpha \beta }$ represents the force per unit length, acting in the ${e}_{\beta }$ direction, on an internal plane perpendicular to the ${e}_{\alpha }$ direction within the plate.

Stress resultant-strain and Moment-Curvature relations

${T}_{\alpha \beta }=\frac{Eh}{\left(1-{\nu }^{2}\right)}\left(\left(1-\nu \right){\gamma }_{\alpha \beta }+\nu {\gamma }_{\lambda \lambda }{\delta }_{\alpha \beta }\right)$      ${M}_{\alpha \beta }=\frac{E{h}^{3}}{12\left(1-{\nu }^{2}\right)}\left(\left(1-\nu \right)\Delta {\kappa }_{\alpha \beta }+\nu \Delta {\kappa }_{\lambda \lambda }{\delta }_{\alpha \beta }\right)$

Equations of motion  reduce to

$\frac{\partial {T}_{\alpha \beta }}{\partial {x}_{\alpha }}+{p}_{\beta }=\rho h\frac{{\partial }^{2}{u}_{\beta }}{d{t}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {V}_{\alpha }}{\partial {x}_{\alpha }}-{T}_{\alpha \beta }\Delta {\kappa }_{\alpha \beta }-{p}_{\alpha }\frac{\partial {u}_{3}}{\partial {x}_{\alpha }}+{p}_{3}=\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}\text{\hspace{0.17em}}$        $\frac{\partial {M}_{\alpha \beta }}{\partial {x}_{\alpha }}-{V}_{\beta }\approx 0$

The second two equations can be combined to eliminate V

$\frac{\partial {M}_{\alpha \beta }}{\partial {x}_{\alpha }\partial {x}_{\beta }}-{T}_{\alpha \beta }{\kappa }_{\alpha \beta }-{p}_{\alpha }\frac{\partial {u}_{3}}{\partial {x}_{\alpha }}+{p}_{3}=\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}$

This result can also be expressed in terms of displacement as

$\frac{E{h}^{3}}{12\left(1-{\nu }^{2}\right)}\frac{{\partial }^{4}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\alpha }\partial {x}_{\beta }\partial {x}_{\beta }}-{T}_{\alpha \beta }\frac{{\partial }^{2}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\beta }}+{p}_{\alpha }\frac{\partial {u}_{3}}{\partial {x}_{\alpha }}+\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}=\text{\hspace{0.17em}}{p}_{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

Edge boundary conditions. The edge of the plate is characterized as described in Section 10.6.1.  Boundary conditions for the transverse displacement ${u}_{3}$, transverse force ${V}_{\alpha }$ and internal moment ${M}_{\alpha \beta }$ are identical to those listed in 10.6.2.   In addition, the in-plane displacements or forces must satisfy the following boundary conditions:

1. On part of the boundary of the plate ${C}_{1}$, one or more components of the in-plane displacement may be prescribed ${u}_{\beta }={u}_{\beta }^{*}$
2. Portions of the boundary of the plate ${C}_{2}$ may be subjected to a prescribed in-plane force per unit length ${P}^{\beta }$. The in-plane forces must then satisfy ${n}_{\alpha }{T}^{\alpha \beta }={P}^{\beta }$

10.6.3 Flat plates with small in-plane and large transverse deflections (von Karman theory)

This version of plate theory is used to model plates that are subjected to substantial loading parallel to the plane of the plate (usually due to loads applied at its boundaries), and also experience substantial out-of-plane displacement.  The theory uses a nonlinear strain measure to account for the in-plane stretching that results from finite transverse displacement and rotation, and includes nonlinear terms associated with the in-plane loading in the equilibrium equations.  The theory can be used to estimate the shape of a buckled plate if the buckling loads are exceeded.

We make the following assumptions

1. The variation of displacements within the plate conforms to the usual approximations of Kirchhoff plate theory;
2. The displacement of the mid-plane of the plate has the form $u={u}_{i}\left({x}_{1},{x}_{2}\right){e}_{i}$: all three displacement components are considered.
3. The in-plane deflections are small, and satisfy $\partial {u}_{\beta }/\partial {x}_{\alpha }<<1$ for all $\alpha ,\beta$; second order terms in these displacement components are ignored in all the strain definitions and also the equilibrium equations.  The out-of-plane displacement is assumed to be small enough to use a linearized measure of curvature.  However, the terms involving products of $\partial {u}_{3}/\partial {x}_{\alpha }$ are retained when computing the strain of the mid-plane of the plate, so that the stretching due to transverse deflection is considered approximately.
1. The in-plane forces are assumed to be much larger than transverse forces.  Nonlinear terms in the equilibrium equations involving in-plane forces are retained; those associated with transverse loading are neglected.

Most of the governing equations of Von-Karman plate theory are identical to those listed in the preceding section.  Nevertheless, the full set of governing equations is give below for convenience.

Kinematics:

The in-plane strain tensor is approximated by ${\gamma }_{\alpha \beta }=\frac{1}{2}\left(\frac{\partial {u}_{\alpha }}{\partial {x}_{\beta }}+\frac{\partial {u}_{\beta }}{\partial {x}_{\alpha }}+\frac{\partial {u}_{3}}{\partial {x}_{\alpha }}\frac{\partial {u}_{3}}{\partial {x}_{\beta }}\right)$.  The additional, nonlinear, term in this expression is the main feature of Von-Karman theory.

The curvature change tensor has components $\Delta {\kappa }_{\alpha \beta }=-\frac{{\partial }^{2}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\beta }}$

The normal vector to the deformed plate can be approximated as ${m}_{3}={e}_{3}-\frac{\partial {u}_{3}}{\partial {x}_{\alpha }}{e}_{\alpha }$

The displacement field in the plate is approximated as $u={u}_{3}{e}_{3}-{x}_{3}\frac{\partial {u}_{3}}{\partial {x}_{\alpha }}{e}_{\alpha }$

The (infinitesimal) strain field in the plate is approximated as ${\epsilon }_{\alpha \beta }={\gamma }_{\alpha \beta }+{x}_{3}\Delta {\kappa }_{\alpha \beta }$

Kinetics

The external force is characterized by the force per unit area $p{e}_{3}$ acting on the surface of the plate.

The transverse stress tensor ${T}_{\alpha \beta }=0$, so the Internal forces are characterized by the in-plane stress resultant tensor ${T}_{\alpha \beta }$, the transverse force tensor ${V}_{\alpha }$ and internal moment tensor ${M}_{\alpha \beta }$.  The physical significance of the components of these tenors is discussed in Sections 10.6.1 and 10.6.2

Stress resultant-strain and Moment-Curvature relations

${T}_{\alpha \beta }=\frac{Eh}{\left(1-{\nu }^{2}\right)}\left(\left(1-\nu \right){\gamma }_{\alpha \beta }+\nu {\gamma }_{\lambda \lambda }{\delta }_{\alpha \beta }\right)$      ${M}_{\alpha \beta }=\frac{E{h}^{3}}{12\left(1-{\nu }^{2}\right)}\left(\left(1-\nu \right)\Delta {\kappa }_{\alpha \beta }+\nu \Delta {\kappa }_{\lambda \lambda }{\delta }_{\alpha \beta }\right)$

Equations of motion  reduce to

$\frac{\partial {T}_{\alpha \beta }}{\partial {x}_{\alpha }}+{p}_{\beta }=\rho h\frac{{\partial }^{2}{u}_{\beta }}{d{t}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {V}_{\alpha }}{\partial {x}_{\alpha }}-{T}_{\alpha \beta }\Delta {\kappa }_{\alpha \beta }-{p}_{\alpha }\frac{\partial {u}_{3}}{\partial {x}_{\alpha }}+{p}_{3}=\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}\text{\hspace{0.17em}}$        $\frac{\partial {M}_{\alpha \beta }}{\partial {x}_{\alpha }}-{V}_{\beta }\approx 0$

where the rotational inertia term has been neglected in the last equation.

Edge boundary conditions:

1. The boundary conditions for transverse displacement ${u}_{3}$ and/or the internal moment  ${M}_{\alpha \beta }$ and transverse force ${V}_{\alpha }$ are identical to those listed in Section 10.6.1
2. The boundary conditions for in-plane displacements ${u}_{\alpha }$ and/or in-plane forces ${T}_{\alpha \beta }$ are identical to those listed in Section 10.6.2.

Alternative forms for the Von-Karman equations: If the plate is in static equilibrium (so the velocity and acceleration of the plate is zero), the Von-Karman equations can be written in a compact form by expressing the in-plane forces ${T}_{\alpha \beta }$ in terms of an Airy stress function $\varphi$, following the procedure outlined for plane elasticity problems in Section 5.2.    As a result, the governing equations can be reduced to a pair of coupled, nonlinear partial differential equations for  $\varphi$ and ${u}_{3}$.  These formulas will not be given here, but can be found, e.g. in Timoshenko and Woinowsky-Krieger, Theory of Plates and Shells,’  McGraw-Hill (1964).

10.6.4 Stretched, flat membrane with small out-of-plane deflections

This is a simplified version of the stretched plate theory outlined in 10.5, which can be used if the plate is so thin that internal bending moments can be neglected.   The problem to be solved is illustrated in the figure.  A membrane’ with Young’s modulus $E$, Poisson’s ratio $\nu$, mass density $\rho$, and thickness h is initially planar and lies in the plane perpendicular to the ${e}_{3}$ direction. The edge of the membrane is subjected to a load per unit length $P={P}_{\alpha }{e}_{\alpha }$ and prevented from moving transverse to the membrane.  A force per unit area $p={p}_{3}{e}_{3}$ acts on the membrane, inducing a small, time dependent, transverse deflection $u={u}_{3}\left({x}_{3}\right){e}_{3}$.

Kinematics:

The in-plane strain tensor is approximated as ${\gamma }_{\alpha \beta }=\left(\partial {u}_{\alpha }/\partial {u}_{\beta }+\partial {u}_{\beta }/\partial {u}_{\alpha }\right)/2$

The displacement field in the plate is approximated as $u={u}_{i}{e}_{i}$.  We assume that ${u}_{3}>>{u}_{\alpha }$.

The curvature of the membrane has components $\Delta {\kappa }_{\alpha \beta }=-\frac{{\partial }^{2}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\beta }}$

The (infinitesimal) strain field in the membrane is approximated as ${\epsilon }_{\alpha \beta }={\gamma }_{\alpha \beta }$

Kinetics

The external force consists of a force per unit area ${p}_{i}{e}_{i}$ acting on the surface of the plate.

The internal forces are characterized by the stress resultant tensor ${T}_{\alpha \beta }$ (the internal moments are neglected).

Stress resultant-strain relations ${T}_{\alpha \beta }=\frac{Eh}{\left(1-{\nu }^{2}\right)}\left(\left(1-\nu \right){\gamma }_{\alpha \beta }+\nu {\gamma }_{\lambda \lambda }{\delta }_{\alpha \beta }\right)$

Equations of motion are approximated as

$\frac{\partial {T}_{\alpha \beta }}{\partial {x}_{\alpha }}\approx 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{T}_{\alpha \beta }\Delta {\kappa }_{\alpha \beta }+{p}_{3}=\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}\text{\hspace{0.17em}}$

The second equation can also be expressed in terms of displacement as

${T}_{\alpha \beta }\frac{{\partial }^{2}{u}_{3}}{\partial {x}_{\alpha }\partial {x}_{\beta }}+{p}_{3}=\rho h\frac{{\partial }^{2}{u}_{3}}{d{t}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

Edge boundary conditions On the edge of the membrane

1. The transverse displacement must satisfy ${u}_{3}=0$
2. The in-plane forces must satisfy ${n}_{\alpha }{T}_{\alpha \beta }={P}_{\beta }$, where n is a unit vector in the $\left({e}_{1},{e}_{2}\right)$ plane perpendicular to the edge of the membrane.

10.6.5 Membrane equations in cylindrical-polar coordinates

In this section, we re-write the governing equations for a stretched membrane in a cylindrical-polar coordinate system, to provide a simple example of the use of general curvilinear coordinates.  We re-consider the membrane described in the preceding section, but now assume that the membrane is circular, with radius R.

HEALTH WARNING: We use polar coordinates $r\equiv {\xi }_{1},\theta \equiv {\xi }_{2}$ as the choice of curvilinear coordinates.  However, all vector and tensor quantities will be expressed as covariant or contravariant components in the natural basis for this coordinate system, not as components in a cylindrical-polar basis of unit vectors $\left\{{e}_{r},{e}_{\theta },{e}_{3}\right\}$.

Coordinate system and kinematic relations

1. Let $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ be a fixed Cartesian basis of mutually perpendicular unit vectors, with ${e}_{3}$ normal to the plane of the undeformed membrane.
2. The position vector of a point in the membrane can be expressed as $\overline{r}=r\mathrm{cos}\theta {e}_{1}+r\mathrm{sin}\theta {e}_{2}$
3. The natural basis vectors and the reciprocal basis (for the undeformed membrane) follow as

$\begin{array}{l}{\overline{m}}_{1}=\frac{\partial \overline{r}}{\partial r}=\mathrm{cos}\theta {e}_{1}+\mathrm{sin}\theta {e}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{m}}_{2}=\frac{\partial \overline{r}}{\partial \theta }=-r\mathrm{sin}\theta {e}_{1}+r\mathrm{cos}\theta {e}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{m}}_{3}={e}_{3}\\ {\overline{m}}^{1}=\mathrm{cos}\theta {e}_{1}+\mathrm{sin}\theta {e}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{m}}^{2}=-\frac{1}{r}\mathrm{sin}\theta {e}_{1}+\frac{1}{r}\mathrm{cos}\theta {e}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{m}}^{3}={e}_{3}\end{array}$

Here the reciprocal basis has simply been written down by inspection $–$ you can readily verify that ${\overline{m}}_{i}\cdot {\overline{m}}^{j}={\delta }_{i}^{j}$.  Note that neither the natural basis vectors nor the reciprocal basis vectors are unit vectors.

1. The Christoffel symbols for the coordinate system and the curvature tensor for the undeformed membrane follow as

$\begin{array}{l}{\overline{\Gamma }}_{\beta \gamma }^{\alpha }={\overline{m}}^{\alpha }\cdot \frac{{\partial }^{2}\overline{r}}{\partial r\partial \theta }⇒{\overline{\Gamma }}_{11}^{1}={\overline{\Gamma }}_{12}^{1}={\overline{\Gamma }}_{21}^{1}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{\Gamma }}_{22}^{1}=-r\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{\Gamma }}_{11}^{2}={\overline{\Gamma }}_{22}^{2}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{\Gamma }}_{12}^{2}={\overline{\Gamma }}_{21}^{2}=\frac{1}{r}\\ {\overline{\kappa }}_{\alpha \beta }=-{\overline{m}}_{3}\cdot \frac{\partial {\overline{m}}_{\alpha }}{\partial {\xi }_{\beta }}=0\end{array}$

1. The position vector of a point in the deformed membrane is $r=r\mathrm{cos}\theta {e}_{1}+r\mathrm{sin}\theta {e}_{2}+{u}_{3}{e}_{3}$
2. The natural basis vectors and reciprocal basis for the deformed membrane follow as

$\begin{array}{l}{m}_{1}=\frac{\partial r}{\partial r}=\mathrm{cos}\theta {e}_{1}+\mathrm{sin}\theta {e}_{2}+\frac{\partial {u}_{3}}{\partial r}{e}_{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{2}=\frac{\partial r}{\partial \theta }=-r\mathrm{sin}\theta {e}_{1}+r\mathrm{cos}\theta {e}_{2}+\frac{\partial {u}_{3}}{\partial \theta }{e}_{3}\\ {m}_{3}={m}_{1}×{m}_{2}/|{m}_{1}×{m}_{2}|\approx -\left(\mathrm{cos}\theta \frac{\partial {u}_{3}}{\partial r}-\mathrm{sin}\theta \frac{1}{r}\frac{\partial {u}_{3}}{\partial \theta }\right){e}_{1}-\left(\mathrm{sin}\theta \frac{\partial {u}_{3}}{\partial r}+\mathrm{cos}\theta \frac{1}{r}\frac{\partial {u}_{3}}{\partial \theta }\right){e}_{2}+{e}_{3}\end{array}$

where terms of order ${\left(\partial {u}_{3}/\partial r\right)}^{2},{\left(\partial {u}_{3}/\partial \theta \right)}^{2}$, etc have been neglected in the expression for ${m}_{3}$.  The reciprocal basis vectors can also be calculated, but are not required in the analysis to follow.

1. For small transverse deflections, the Christoffel symbols associated with the deformed membrane can be approximated using those for the undeformed membrane.  The curvature tensor for the deformed membrane has covariant components

${\kappa }_{11}=-\frac{{\partial }^{2}{u}_{3}}{\partial {r}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\kappa }_{12}={\kappa }_{21}=\frac{1}{r}\frac{\partial {u}_{3}}{\partial \theta }-\frac{{\partial }^{2}{u}_{3}}{\partial r\partial \theta }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\kappa }_{22}=-r\frac{\partial {u}_{3}}{\partial r}-\frac{{\partial }^{2}{u}_{3}}{\partial {\theta }^{2}}$

Notice that the curvature components all have different units $–$ this is because the basis vectors themselves have units. It is easy to check that the terms in the dyadic product ${\kappa }_{\alpha \beta }{m}^{\alpha }\otimes {m}^{\beta }$ all have correct units.

Equations of Motion: The general equations of motion for a shell are

$\begin{array}{l}\frac{\partial {T}^{\alpha \beta }}{\partial {\xi }_{\alpha }}+{T}^{\alpha \beta }{\Gamma }_{\alpha \gamma }^{\gamma }+{T}^{\alpha \gamma }{\Gamma }_{\gamma \alpha }^{\beta }+{V}^{\alpha }{\kappa }_{\alpha }^{\beta }+{p}^{\beta }=\rho h{a}^{\beta }\\ \frac{\partial {V}^{\alpha }}{\partial {\xi }_{\alpha }}+{V}^{\alpha }{\Gamma }_{\alpha \beta }^{\beta }-{T}^{\alpha \beta }{\kappa }_{\alpha \beta }+{p}^{3}=\rho h{a}^{3}\end{array}$ $\begin{array}{l}\frac{\partial {M}^{\alpha \beta }}{\partial {\xi }_{\alpha }}+{M}^{\alpha \beta }{\Gamma }_{\alpha \gamma }^{\gamma }+{M}^{\alpha \gamma }{\Gamma }_{\gamma \alpha }^{\beta }-{V}^{\beta }+{q}^{\beta }=\frac{\rho {h}^{3}}{12}{\stackrel{¨}{\mu }}^{\beta }\\ {T}^{12}-{T}^{21}+{M}^{\alpha 1}{\kappa }_{\alpha }^{2}-{M}^{\alpha 2}{\kappa }_{\alpha }^{1}=0\end{array}$

We proceed to simplify these for a flat membrane.

1. No external moments ${q}^{\beta }$ act on the membrane, and the membrane thickness is assumed to be so small that the internal moments ${M}^{\alpha \beta }$ can be neglected. We may also assume $\rho {h}^{3}\approx 0$.   The equations of motion in the right-hand column then show that transverse forces ${V}^{\alpha }$ must vanish, and that the in-plane forces ${T}^{\alpha \beta }$ are symmetric ${T}^{12}={T}^{21}$.
2. Substituting for the Christoffel symbols and curvature components into the remaining equations of motion and recalling that in-plane forces ${p}^{\beta }=0$, we find that the three remaining equilibrium equations reduce to

$\begin{array}{l}\frac{\partial {T}^{11}}{\partial r}+\frac{\partial {T}^{21}}{\partial \theta }+\frac{{T}^{11}}{r}-{T}^{22}r=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {T}^{22}}{\partial \theta }+\frac{\partial {T}^{12}}{\partial r}+3\frac{{T}^{12}}{r}=0\\ {T}^{11}\frac{{\partial }^{2}{u}_{3}}{\partial {r}^{2}}+2{T}^{12}\left(\frac{{\partial }^{2}{u}_{3}}{\partial r\partial \theta }-\frac{1}{r}\frac{\partial {u}_{3}}{\partial \theta }\right)+{T}^{22}\left(r\frac{\partial {u}_{3}}{\partial r}+\frac{{\partial }^{2}{u}_{3}}{\partial {\theta }^{2}}\right)+{p}^{3}=\rho h\frac{{\partial }^{2}{u}_{3}}{\partial {t}^{2}}\end{array}$

Boundary conditions:

1. The transverse displacement must satisfy ${u}_{3}=0$ on the edge of the membrane at r=R.
2. The in-plane forces must satisfy ${T}^{1\beta }={P}^{\beta }$, where $P={P}^{\beta }{\overline{m}}_{\beta }$ is the force per unit length acting on the edge of the membrane at r=R.

Special case: membrane subjected to uniform biaxial in-plane loading: If the membrane is subjected to a uniform radial force per unit length $P={T}_{0}{\overline{m}}_{1}$ acting on its edge at r=R, the first two equations of motion and the boundary conditions are satisfied by ${T}^{11}={T}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}^{22}={T}_{0}/{r}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}^{12}=0$.  This corresponds to a state of uniform biaxial tension in the membrane.  The equation of motion for the transverse deflection reduces to

${T}_{0}\left(\frac{{\partial }^{2}{u}_{3}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {u}_{3}}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}{u}_{3}}{\partial {\theta }^{2}}\right)+{p}^{3}=\rho h\frac{{\partial }^{2}{u}_{3}}{\partial {t}^{2}}$