Chapter 10

Approximate theories for solids with special shapes:

rods, beams, membranes, plates and shells

10.5 Motion and Deformation of thin shells $–$ General theory

The figure illustrates the problem to be solved.   The solid of interest is a shell with uniform thickness h.  The shell’s thickness is assumed to be much smaller than any relevant in-plane dimension.   The exterior surface of the shell is subjected to a prescribed distribution of traction, while the edge of the shell may either be supported so as to constrain its motion, or may be subjected to prescribed forces.  Our objective is to calculate the internal forces in the shell, and to compute its deformed shape.

10.5.1 Coordinate systems and variables characterizing deformation of shells

To specify the position of a point on the mid-plane of the undeformed shell, we introduce a convenient curvilinear coordinate system $\left({\xi }_{1},{\xi }_{2}\right)$ (examples include cylindrical or spherical polar coordinates).  Note that ${\xi }_{\alpha }$ need not necessarily be distances along the surface: for example, for a cylindrical shell, we would use the axial distance z and the angle $\theta$ as the coordinate system.

The position vector of a material particle on the mid-section of the initial shell is denoted by $\overline{r}\left({\xi }_{1},{\xi }_{2}\right)$

To characterize the orientation of an arbitrary point in the undeformed shell, we introduce three basis vectors $\left({\overline{m}}_{1},{\overline{m}}_{2},{\overline{m}}_{3}\right)$, with

${\overline{m}}_{\alpha }=\frac{\partial \overline{r}}{\partial {\xi }_{\alpha }}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{m}}_{3}=\frac{{\overline{m}}_{1}×{\overline{m}}_{2}}{|{\overline{m}}_{1}×{\overline{m}}_{2}|}$.

Thus, ${\overline{m}}_{\alpha }$ are tangent to the coordinate lines ${\xi }_{\alpha }$ in the undeformed shell, and ${\overline{m}}_{3}$ is a unit vector perpendicular to the mid-section of the shell.  This basis is called the covariant basis or natural basis for the coordinate system.  Note that the basis vectors ${\overline{m}}_{\alpha }$ are not unit vectors, and are not, in general, orthogonal.

Because $\left({\overline{m}}_{1},{\overline{m}}_{2},{\overline{m}}_{3}\right)$ are not orthogonal, it is convenient to introduce a second set of basis vectors $\left({\overline{m}}^{1},{\overline{m}}^{2},{\overline{m}}^{3}\right)$ defined so that

${\overline{m}}^{i}\cdot {\overline{m}}_{j}={\delta }_{j}^{i}$

where ${\delta }_{j}^{i}$ is the Kronecker delta symbol (the index i has been raised to match the indices on the basis vectors), i.e. ${\delta }_{j}^{i}=1$ for i=j and zero otherwise.  This second triad of vectors is called the contravariant basis or reciprocal basis for the coordinate system.  The contravariant basis vectors can be constructed by taking cross products of the covariant basis vectors, as follows

${\overline{m}}^{1}=\beta {\overline{m}}_{2}×{\overline{m}}_{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{m}}^{2}=\beta {\overline{m}}_{3}×{\overline{m}}_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{m}}^{3}={\overline{m}}_{3}$

where $\beta =1/{\overline{m}}_{1}\cdot \left({\overline{m}}_{2}×{\overline{m}}_{3}\right)=1/|{\overline{m}}_{1}×{\overline{m}}_{2}|$.

The position vector of an arbitrary point in the undeformed shell can be expressed as $x=\overline{r}\left({\xi }_{1},{\xi }_{2}\right)+{x}_{3}{\overline{m}}_{3}$, where ${x}_{3}$ is the perpendicular distance of the material particle from the mid-section of the shell.

After deformation, the mid-section of the shell is deformed to another smooth surface.  The point that lies at $x=\overline{r}\left({\xi }_{1},{\xi }_{2}\right)$ on the mid-section of the undeformed shell moves to a new position $y=r\left({\xi }_{1},{\xi }_{2}\right)$ after deformation.

To characterize the orientation of the deformed shell, we introduce three basis vectors $\left({m}_{1},{m}_{2},{m}_{3}\right)$, with ${m}_{\alpha }=\frac{\partial r}{\partial {\xi }_{\alpha }}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{3}={m}_{1}×{m}_{2}/\left(|{m}_{1}×{m}_{2}|\right)$.  Now, ${m}_{\alpha }$ are tangent to the coordinate lines ${\xi }_{\alpha }$ in the deformed shell, and ${m}_{3}$ is a unit vector perpendicular to the mid-section of the deformed shell.  We can introduce a reciprocal basis $\left({m}^{1},{m}^{2},{m}^{3}\right)$ in exactly the same way as for the undeformed shell.

A few special vectors and tensors, such as the angular velocity of the shell, and the internal stress couple in the shell are most conveniently expressed in terms of vectors ${m}_{3}×{m}_{\alpha }$ or ${m}_{3}×{m}^{\alpha }$.  Special symbols will not be introduced for these vectors; they will always be written out as a cross product.

10.5.2 Vectors and tensor components in non-orthogonal bases: Covariant and Contravariant components

In this section we introduce some additional notation that helps deal with the complicated sets of basis vectors that characterize the deformation of a shell.

Vectors can be expressed as linear combinations of some subset of the twelve possible basis vectors $\left({\overline{m}}_{1},{\overline{m}}_{2},{\overline{m}}_{3}\right)$, $\left({\overline{m}}^{1},{\overline{m}}^{2},{\overline{m}}^{3}\right)$, $\left({m}_{1},{m}_{2},{m}_{3}\right)$ or $\left({m}^{1},{m}^{2},{m}^{3}\right)$.  For example, we can write an arbitrary vector a as

$a={\overline{a}}^{i}{\overline{m}}_{i}={\overline{a}}_{i}{\overline{m}}^{i}={a}^{i}{m}_{i}={a}_{i}{m}^{i}$

Here, the coefficients ${\overline{a}}^{i}$ are called the contravariant components of a in $\left({\overline{m}}_{1},{\overline{m}}_{2},{\overline{m}}_{3}\right)$, and ${\overline{a}}_{i}$ are called the covariant components of a.  Note that the contravariant components are coefficients of the covariant basis vectors, and vice-versa..  The reason for this confusing terminology is given below.  Note also that the components do not in general have the same units as the vector, because the basis vectors may have length dimensions.

The various components of a can be expressed as

${\overline{a}}^{i}={\overline{m}}^{i}\cdot a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{a}}_{i}={\overline{m}}_{i}\cdot a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}^{i}={m}^{i}\cdot a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{i}={m}_{i}\cdot a$

To see the first result, take dot products of $a={\overline{a}}^{i}{\overline{m}}_{i}$ with ${m}^{j}$ and recall that ${\overline{m}}^{i}\cdot {\overline{m}}_{j}={\delta }_{j}^{i}$. The contravariant’ and covariant’ terms assigned to ${\overline{a}}^{i}$ and ${\overline{a}}_{i}$ refer to the fact that they represent projections of the vector a onto the contravariant and covariant basis vectors, respectively.  The raised and lowered indices for vector components follow the same convention: raised indices indicate contravariant components, while lowered indices represent covariant components.

Tensors can also be expressed as sums of nine dyadic products of various combinations of basis vectors.  For example, if S is a tensor we could write

$S={S}^{ij}{m}_{i}\otimes {m}_{j}={S}_{\cdot j}^{i}{m}_{i}\otimes {m}^{j}={S}_{j}^{\cdot i}{m}^{j}\otimes {m}_{i}={S}_{ij}{m}^{i}\otimes {m}^{j}$

or equivalent expressions in terms of ${\overline{m}}_{i}$ and/or ${\overline{m}}^{j}$.  Here ${S}^{ij},{S}_{ij}$ are called contravariant and covariant components of S, respectively, while ${S}_{.j}^{i},{S}_{j}^{.i}$ are called mixed tensor components.  The dot that appears before the indices in the mixed tensors is introduced to identify whether the index should be associated with the first or second basis vector in the dyadic product (the dot appears before the index associated with the second basis vector).  For symmetric tensors, the dot can be dropped.

The various components of S can be regarded as projections of the tensor onto the contravariant or covariant basis vectors, as ${S}_{ij}={m}_{i}\cdot S\cdot {m}_{j}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{S}^{ij}={m}^{i}\cdot S\cdot {m}^{j}$, with similar results for the mixed components.

Once again, it is important to note that the components ${S}_{ij}$ do not have a convenient physical interpretation.  In general the components do not even have the same units as the tensor itself, because the basis vectors themselves have units.

The various sets of components can be related by defining the components of the fundamental tensor or metric tensor g as follows

${g}_{ij}={m}_{i}\cdot {m}_{j}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{g}^{ij}={m}^{i}\cdot {m}^{j}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{g}_{j}^{i}={m}^{i}\cdot {m}_{j}={\delta }_{j}^{i}$

We can define ${\overline{g}}_{ij}$ in terms of ${\overline{m}}_{i}$ and/or ${\overline{m}}^{j}$ in the same way.  With these definitions, we see that the covariant and contravariant basis vectors are related by

${m}^{i}={g}^{ij}{m}_{j}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{i}={g}_{ij}{m}^{j}$

The metric tensor can be expressed as the sum of three dyadic products $g={m}_{i}\otimes {m}^{i}$, with a similar expression for $\overline{g}$.  From this expression we see that

$gg=\left({m}_{i}\otimes {m}^{i}\right)\left({m}_{j}\otimes {m}^{j}\right)={m}_{i}\otimes {\delta }_{i}^{j}{m}^{j}={m}_{i}\otimes {m}^{i}=g$

g is therefore its own inverse $–$ it is a representation of the identity tensor.

10.5.3 Additional Deformation Measures and Kinematic Relations

An infinitesimal line element $d\overline{r}$ that lies in the mid-section of the undeformed plate can be expressed in terms of infinitesimal changes in the coordinates $d{\xi }_{\alpha }$ as

$d\overline{r}={\overline{m}}_{\alpha }d{\xi }_{\alpha }$

The length ds of $d\overline{r}$ can be computed as

$d{\overline{s}}^{2}=d\overline{r}\cdot d\overline{r}={\overline{m}}_{\alpha }\cdot {\overline{m}}_{\beta }d{\xi }_{\alpha }d{\xi }_{\beta }={\overline{g}}_{\alpha \beta }d{\xi }_{\alpha }d{\xi }_{\beta }$

This expression is known as the first fundamental form for the surface.  A similar expression can be constructed for the deformed surface.

The variation of the normal vectors ${\overline{m}}_{3}$ and ${m}_{3}$ with position in the mid-plane of the shell play a particularly important role in describing the shape and deformation of the shell, because they characterize its curvature and bending.  To quantify this variation, let ${\overline{m}}_{3}$ and ${\overline{m}}_{3}+d{\overline{m}}_{3}$ be the vectors normal to the surface at positions $\overline{r}$ and $\overline{r}+d\overline{r}$ in the undeformed shell, with a similar notation for the deformed shell. We introduce symmetric curvature tensors $\overline{\kappa }$ and $\kappa$ that satisfy

$d{\overline{m}}_{3}=\overline{\kappa }d\overline{r}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{m}_{3}=\kappa dr$

The curvatures $\overline{\kappa }$ and $\kappa$ are called surface tensors, because they transform like tensors under changes of surface coordinates.

The curvature components can be expressed in terms of their covariant, contravariant or mixed components.   This can be used to deduce expressions such as

$\begin{array}{l}d{\overline{m}}_{3}=\left({\overline{\kappa }}_{\alpha \gamma }{\overline{m}}^{\alpha }\otimes {\overline{m}}^{\gamma }\right)\cdot \left({\overline{m}}_{\beta }d{\xi }_{\beta }\right)={\overline{\kappa }}_{\alpha \beta }d{\xi }_{\beta }{\overline{m}}^{\alpha }\\ d{\overline{m}}_{3}=\left({\overline{\kappa }}_{\gamma }^{\alpha }{\overline{m}}_{\alpha }\otimes {\overline{m}}^{\gamma }\right)\cdot \left({\overline{m}}_{\beta }d{\xi }_{\beta }\right)={\overline{\kappa }}_{\beta }^{\alpha }d{\xi }_{\beta }{\overline{m}}_{\alpha }\end{array}$

The curvature components can be calculated from the position vector of the mid-plane of the shell, using any of the following expressions

$\begin{array}{l}{\overline{\kappa }}_{\alpha \beta }={\overline{m}}_{\alpha }\cdot \frac{\partial {\overline{m}}_{3}}{\partial {\xi }_{\beta }}=\frac{\partial \overline{r}}{\partial {\xi }_{\alpha }}\cdot \frac{\partial }{\partial {\xi }_{\beta }}\left(\frac{1}{\lambda }\frac{\partial \overline{r}}{\partial {\xi }_{1}}×\text{\hspace{0.17em}}\frac{\partial \overline{r}}{\partial {\xi }_{2}}\right)=-\left(\frac{1}{\lambda }\frac{\partial \overline{r}}{\partial {\xi }_{1}}×\text{\hspace{0.17em}}\frac{\partial \overline{r}}{\partial {\xi }_{2}}\right)\cdot \left(\frac{{\partial }^{2}\overline{r}}{\partial {\xi }_{\alpha }\partial {\xi }_{\beta }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-{\overline{m}}_{3}\cdot \frac{\partial {\overline{m}}_{\alpha }}{\partial {\xi }_{\beta }}=-{\overline{m}}_{3}\cdot \frac{\partial {\overline{m}}_{\beta }}{\partial {\xi }_{\alpha }}\end{array}$

where $\lambda =|\left(\partial \overline{r}/\partial {\xi }_{1}\right)×\left(\partial \overline{r}/\partial {\xi }_{2}\right)|$. The mixed components follow as ${\kappa }_{\beta }^{\alpha }={g}^{\alpha \gamma }{\kappa }_{\gamma \beta }$, where ${g}^{\alpha \gamma }$ are the components of the metric tensor defined in Sect 10.5.2.

The magnitude of the curvature of a shell is quantified by principal curvatures ${\overline{\kappa }}_{1},{\overline{\kappa }}_{2}$ - these are simply the principal values of $\kappa$. The mean curvature $\left({\kappa }_{1}+{\kappa }_{2}\right)/2$, and Gaussian curvature ${\kappa }_{1}{\kappa }_{2}$ are also used.

We will also need to calculate the variation of the remaining basis vectors with position in the surface.  These are quantified by Christoffel symbols of the second kind  ${\Gamma }_{\alpha \beta }^{i}$ which satisfy

$d{m}_{i}={\Gamma }_{i\alpha }^{k}{m}_{k}d{\xi }_{\alpha }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{m}^{i}=-{\Gamma }_{k\alpha }^{i}{m}^{k}d{\xi }_{\alpha }$

The Christoffel symbols are functions of position on the surface, and can be related to the position vector of the mid-plane of the shell and its curvature components as

${\Gamma }_{\beta \gamma }^{\alpha }={m}^{\alpha }\cdot \frac{{\partial }^{2}r}{\partial {\xi }_{\beta }\partial {\xi }_{\gamma }}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Gamma }_{\alpha \beta }^{3}=-{\kappa }_{\alpha \beta }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Gamma }_{3\beta }^{\alpha }={\kappa }_{\beta }^{\alpha }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Gamma }_{3\alpha }^{3}=0$

Some relationships between the time derivatives of these various kinematic quantities are also needed in subsequent calculations.  The rate of change in shape of the shell can be characterized by the velocity of its middle surface $v\left({\xi }_{1},{\xi }_{2}\right)=dr/dt$.  The velocity vector can be described as components in any of the various bases: the representation $v={v}_{i}{m}^{i}$ is particularly useful.

The time derivatives of the basis vectors ${m}_{\alpha }$ are a convenient way to characterize the rate of change of bending of the shell.   These are related to the velocity of the shell’s mid-plane by

$\begin{array}{l}\frac{d{m}_{\alpha }}{dt}=\frac{d}{dt}\left(\frac{dr}{d{\xi }_{\alpha }}\right)=\frac{dv}{d{\xi }_{\alpha }}=\frac{\partial {v}^{i}}{\partial {\xi }_{\alpha }}{m}_{i}+{v}^{i}\frac{d{m}_{i}}{d{\xi }_{\alpha }}=\frac{\partial {v}^{i}}{\partial {\xi }_{\alpha }}{m}_{i}+{v}^{i}{\Gamma }_{i\alpha }^{k}{m}_{k}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{\partial {v}^{\beta }}{\partial {\xi }_{\alpha }}{m}_{\beta }+{v}^{\beta }{\Gamma }_{\beta \alpha }^{\lambda }{m}_{\lambda }+{v}^{3}{\kappa }_{\alpha }^{\beta }{m}_{\beta }+\left(\frac{\partial {v}^{3}}{\partial {\xi }_{\alpha }}-{v}^{\beta }{\kappa }_{\beta \alpha }\right){m}_{3}\\ \text{\hspace{0.17em}}\frac{d{m}_{\alpha }}{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{\partial {v}_{i}}{\partial {\xi }_{\alpha }}{m}^{i}+{v}_{i}\frac{d{m}^{i}}{d{\xi }_{\alpha }}=\frac{\partial {v}_{i}}{\partial {\xi }_{\alpha }}{m}^{i}-{v}_{i}{\Gamma }_{k\alpha }^{i}{m}^{k}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{\partial {v}_{\beta }}{\partial {\xi }_{\alpha }}{m}^{\beta }-{v}_{\beta }{\Gamma }_{\lambda \alpha }^{\beta }{m}^{\lambda }+{v}_{3}{\kappa }_{\beta \alpha }{m}^{\beta }+\left(\frac{\partial {v}_{3}}{\partial {\xi }_{\alpha }}-{v}_{\beta }{\kappa }_{\alpha }^{\beta }\right){m}^{3}\end{array}$

We will also need to calculate the time derivative of the vector normal to the mid-plane of the shell ${\stackrel{˙}{m}}_{3}$.  Since ${m}_{3}$ is a unit vector, its time derivative can be quantified by an angular velocity vector $\omega$, defined so that

$\frac{d{m}_{3}}{dt}=\omega ×{m}_{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \cdot {m}_{3}=0$

The components of $\omega$ can be related to $d{m}_{\alpha }/dt$ as

${m}_{3}×\left(\omega ×{m}_{3}\right)=\omega ={m}_{3}×\frac{d{m}_{3}}{dt}={m}_{3}×\frac{\left({\stackrel{˙}{m}}_{1}×{m}_{2}+{m}_{1}×{\stackrel{˙}{m}}_{2}\right)}{|{m}_{1}×{m}_{2}|}=\frac{-\left({m}_{3}\cdot {\stackrel{˙}{m}}_{1}\right){m}_{2}+\left({m}_{3}\cdot {\stackrel{˙}{m}}_{2}\right){m}_{1}}{|{m}_{1}×{m}_{2}|}$

Recalling that ${m}_{2}×{m}_{3}/\left(|{m}_{1}×{m}_{2}|\text{\hspace{0.17em}}\right)={m}^{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{3}×{m}_{1}/\left(|{m}_{1}×{m}_{2}|\right)=\text{\hspace{0.17em}}{m}^{2}$, we see also that

$\frac{d{m}_{3}}{dt}=\omega ×{m}_{3}=-\left({m}_{3}\cdot {\stackrel{˙}{m}}_{1}\right){m}^{1}-\left({m}_{3}\cdot {\stackrel{˙}{m}}_{2}\right){m}^{2}=-\left({m}_{3}\cdot {\stackrel{˙}{m}}_{\alpha }\right){m}^{\alpha }$

Finally we may write this as

$\frac{d{m}_{3}}{dt}={\stackrel{˙}{\mu }}_{\alpha }{m}^{\alpha }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˙}{\mu }}_{\alpha }=-{m}_{3}\cdot {\stackrel{˙}{m}}_{\alpha }=-\left(\frac{\partial {v}_{3}}{\partial {\xi }_{\alpha }}-{v}_{\beta }{\kappa }_{\alpha }^{\beta }\right)$

The components ${\stackrel{˙}{\mu }}_{\alpha }$ can also be regarded as the components of the angular velocity vector $\omega$ in a basis $\left\{{m}_{3}×{m}^{1},{m}_{3}×{m}^{2}\right\}$ in the sense that

$\omega ={\stackrel{˙}{\mu }}_{\alpha }{m}_{3}×{m}^{\alpha }$

The time derivative of the curvature tensor $\stackrel{˙}{\kappa }$, is related to ${\stackrel{˙}{m}}_{3}$ by

$d{\stackrel{˙}{m}}_{3}={\stackrel{˙}{\kappa }}^{\alpha }{}_{\beta }d{\xi }_{\beta }{m}_{\alpha }$

Note that

$d{\stackrel{˙}{m}}_{3}=\left(\frac{\partial {\stackrel{˙}{\mu }}_{\alpha }}{\partial {\xi }_{\beta }}{m}^{\alpha }+{\stackrel{˙}{\mu }}_{\alpha }\frac{\partial {m}^{\alpha }}{\partial {\xi }_{\beta }}\right)d{\xi }_{\beta }=\left(\frac{\partial {\stackrel{˙}{\mu }}_{\alpha }}{\partial {\xi }_{\beta }}{m}^{\alpha }-{\stackrel{˙}{\mu }}_{\alpha }{\Gamma }_{\lambda \beta }^{\alpha }{m}^{\lambda }\right)d{\xi }_{\beta }$

The components of the time derivative of curvature can therefore be expressed in terms of ${\stackrel{˙}{\mu }}_{\alpha }$ as

${\stackrel{˙}{\kappa }}_{\beta }^{\alpha }{m}_{\alpha }={\stackrel{˙}{\kappa }}_{\alpha \beta }{m}^{\alpha }=\left(\frac{\partial {\stackrel{˙}{\mu }}_{\alpha }}{\partial {\xi }_{\beta }}{m}^{\alpha }-{\stackrel{˙}{\mu }}_{\lambda }{\Gamma }_{\alpha \beta }^{\lambda }{m}^{\alpha }\right)$

It is important to note that ${\stackrel{˙}{\kappa }}_{\alpha \beta }$ are not equal to the time derivatives of the curvature components.

We will also need to characterize the linear and angular acceleration of the shell.  The linear acceleration can be quantified by the acceleration of the mid-plane $a={d}^{2}r/d{t}^{2}$

The angular acceleration of the shell can be characterized by the angular acceleration of the normal to its mid-plane, $\alpha =d\omega /dt$.  The angular acceleration can be related to the acceleration of the mid-plane of the shell as follows

$\alpha =\frac{d\omega }{dt}={m}_{3}×\frac{{d}^{2}{m}_{3}}{d{t}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[2\left({m}_{3}\cdot {\stackrel{˙}{m}}_{\alpha }\right)\left({m}^{\alpha }\cdot {\stackrel{˙}{m}}_{\beta }\right)-{m}_{3}\cdot \frac{da}{d{\xi }_{\beta }}\right]{m}_{3}×{m}^{\beta }$

where we have used ${\stackrel{˙}{m}}_{3}=-\left({m}_{3}\cdot {\stackrel{˙}{m}}_{\alpha }\right){m}^{\alpha }$ and noted that  ${m}^{\alpha }\cdot {m}_{\beta }={\delta }_{\beta }^{\alpha }$ $⇒{\stackrel{˙}{m}}^{\alpha }\cdot {m}_{\beta }+{m}^{\alpha }\cdot {\stackrel{˙}{m}}_{\beta }=0$ to obtain the final result.

It is convenient to express $\alpha$ in the form $\alpha ={\stackrel{¨}{\mu }}_{\beta }{m}_{3}×{m}^{\beta }={\stackrel{¨}{\mu }}^{\beta }{m}_{3}×{m}_{\beta }$, where ${\stackrel{¨}{\mu }}^{\beta }={g}^{\beta \alpha }{\stackrel{¨}{\mu }}_{\alpha }$ and the ${\stackrel{¨}{\mu }}_{\beta }$ can be related to the velocity and acceleration of the mid-plane of the shell as follows

$\begin{array}{l}{\stackrel{¨}{\mu }}_{\beta }=2\left({m}_{3}\cdot {\stackrel{˙}{m}}_{\alpha }\right)\left({m}^{\alpha }\cdot {\stackrel{˙}{m}}_{\beta }\right)-{m}_{3}\cdot \frac{da}{d{\xi }_{\beta }}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2\left(\frac{\partial {v}_{3}}{\partial {\xi }_{\alpha }}-{v}^{\lambda }{\kappa }_{\alpha \lambda }\right)\left(\frac{\partial {v}^{\alpha }}{\partial {\xi }_{\beta }}+{v}^{\rho }{\Gamma }_{\beta \rho }^{\alpha }+{v}_{3}{\kappa }_{\beta }^{\alpha }\right)-\left(\frac{\partial {a}_{3}}{\partial {\xi }_{\beta }}-{a}^{\alpha }{\kappa }_{\alpha \beta }\right)\end{array}$

These results show that

$\begin{array}{l}\frac{{d}^{2}{m}_{3}}{d{t}^{2}}=\frac{d}{dt}\left(\omega ×{m}_{3}\right)=\alpha ×{m}_{3}+\omega ×\left(\omega ×{m}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\stackrel{¨}{\mu }}^{\beta }{m}_{\beta }-{\stackrel{˙}{\mu }}_{\alpha }{\stackrel{˙}{\mu }}^{\alpha }{m}_{3}\end{array}$

where we have used $\omega ×\left(\omega ×{m}_{3}\right)=\left(\omega \cdot {m}_{3}\right)\omega -\left(\omega \cdot \omega \right){m}_{3}$ and noted  $\left(\omega \cdot {m}_{3}\right)=0$ to obtain the second line.

HEALTH WARNING: The sign convention used to characterize the curvature of a shell can be confusing.  In the convention used here, a convex surface has positive curvature. For example, a spherical shell with coordinate system chosen so that ${m}_{3}$ points radially out of the sphere would have two equal positive principal curvatures. The mathematical analysis of curved surfaces usually uses the opposite sign convention for curvature, and a few texts on shell theory use curvature measures with the opposite sign to the one used here.

10.5.4 Approximating the displacement and velocity field

The position vector of a material point in the shell before deformation can be expressed as $x=\overline{r}\left({\xi }_{1},{\xi }_{2}\right)+{x}_{3}{\overline{m}}_{3}$, where ${x}_{3}$ is the distance of the material particle from the mid-section of the shell.

After deformation of the material point that has coordinates $\left({\xi }_{1},{\xi }_{2},{x}_{3}\right)$ in the undeformed shell moves to a new position, which can be expressed as

$y\left({\xi }_{\alpha },{x}_{3}\right)=r\left({\xi }_{\alpha }\right)+{\eta }^{i}\left({\xi }_{\alpha },{x}_{k}\right){m}_{i}\left({\xi }_{\alpha }\right)$

where ${\eta }^{i}\left({\xi }_{\alpha },0\right)=0$. This is a completely general expression.   We now introduce a series of approximations that are based on the assumptions that

1. The shell is thin compared with its in-plane dimensions;
2. The principal radii of curvature of the shell (both before and after deformation) are much larger than the characteristic dimension of its cross section;

With this in mind, we assume that ${\eta }^{i}\left({\xi }_{\alpha },{x}_{3}\right)$  can be approximated by a function of the form

${\eta }^{i}\left({\xi }_{\alpha },{x}_{3}\right)=\left({\delta }_{3}^{i}+{f}_{3}^{i}\left({\xi }_{\alpha }\right)\right){x}_{3}$

where ${f}_{3}^{i}$ can be regarded as the first term in a Taylor expansion of ${\eta }^{i}$ with respect to ${x}_{3}$.  Note that  ${f}_{3}^{\alpha }$ represents transverse shear deformation of the shell, while ${f}_{3}^{3}$ quantifies the through-thickness stretching.

Several versions of plate theory exist, which use different approximations for the shear deformation.  Here, we will present only simplest version, known as Kirchhoff shell theory,  which is to assume that ${f}_{3}^{i}=0$.  This implies that material fibers that are perpendicular to the mid-plane of the shell remain perpendicular to the mid-plane of the deformed shell, and the shell does not change its thickness. This reduces the displacement field to

$y\left({\xi }_{\alpha },{x}_{3}\right)=r\left({\xi }_{\alpha }\right)+{x}_{3}{m}_{3}\left({\xi }_{\alpha }\right)$

The velocity field can be approximated as

$\frac{d}{dt}y\left({\xi }_{\alpha },{x}_{3}\right)=v+{x}_{3}{\stackrel{˙}{m}}_{3}$

while the acceleration is

$\frac{{d}^{2}}{d{t}^{2}}y\left({\xi }_{\alpha },{x}_{3}\right)=a+{x}_{3}\frac{{d}^{2}{m}_{3}}{d{t}^{2}}=a+{x}_{3}\left[\alpha ×{m}_{3}+\omega ×\left(\omega ×{m}_{3}\right)\right]$

where $\alpha$ and $\omega$ denote the angular acceleration and angular velocity of the unit vector normal to the mid-plane of the plate.

HEALTH WARNING: In addition to using this approximation to the displacement and velocity field, Kirchhoff shell theory assumes that the transverse stress ${\sigma }^{33}$ vanishes in the shell.  Strictly speaking, this is inconsistent with the deformation.  A more rigorous approach would be to introduce a uniform transverse strain ${f}_{3}^{3}$,  which could be calculated as part of the solution.  However, this approach yields results that are essentially indistinguishable from the plane-stress approximation.

The deformation gradient can be approximated as

$\begin{array}{l}F=\left(g+{x}_{3}\kappa \right)\left({m}_{\alpha }\otimes {\overline{m}}^{\alpha }\right){\left(}^{\overline{g}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx {m}_{\alpha }\otimes {\overline{m}}^{\alpha }+{x}_{3}\left({\kappa }_{\beta }^{\alpha }-{\overline{\kappa }}_{\beta }^{\alpha }\right){m}_{\alpha }\otimes {\overline{m}}^{\beta }+{m}_{3}\otimes {\overline{m}}^{3}\end{array}$

where $\overline{g}={\overline{m}}_{i}\otimes {\overline{m}}^{i}$ and $g={m}_{i}\otimes {m}^{i}$ are the metric tensors for shell before and after deformation, and $\overline{\kappa }={\overline{\kappa }}_{\beta }^{\alpha }{\overline{m}}_{\alpha }\otimes {\overline{m}}^{\beta }$ and $\kappa ={\kappa }_{\beta }^{\alpha }{m}_{\alpha }\otimes {m}^{\beta }$ are the curvature tensors for the mid-surface of the shell before and after deformation, respectively.  The three terms in the second formula for F can be interpreted as (i) the effects of in-plane stretching of the shell; (ii) the effects of bending; and (iii) the effects of a change in the shell’s thickness.

Derivation  By definition, the deformation gradient relates infinitesimal line elements in the shell before $\left(dx\right)$ and after $\left(dy\right)$ deformation by $dy=Fdx$.  We wish to construct a tensor with these properties.

1. An infinitesimal line element in the deformed shell can be expressed in terms of a small change in coordinates

$dy=\frac{dr}{d{\xi }_{\alpha }}d{\xi }_{\alpha }+{x}_{3}\frac{d{m}_{3}}{d{\xi }_{\alpha }}d{\xi }_{\alpha }+d{x}_{3}{m}_{3}=\frac{dr}{d{\xi }_{\alpha }}d{\xi }_{\alpha }+{x}_{3}{\overline{\kappa }}_{\beta }^{\alpha }d{\xi }_{\beta }{m}_{\alpha }+d{x}_{3}{m}_{3}$

where we have used $\frac{\partial {m}_{3}}{\partial {\xi }_{\beta }}d{\xi }_{\beta }={\kappa }_{\beta }^{\alpha }d{\xi }_{\beta }{m}_{\alpha }$

1. This expression can be rearranged into the form

$\begin{array}{l}dy=\left({\delta }_{\beta }^{\alpha }{m}_{\alpha }\otimes {m}^{\beta }+{m}_{3}\otimes {m}^{3}+{x}_{3}{\kappa }_{\beta }^{\alpha }{m}_{\alpha }\otimes {m}^{\beta }\right)\cdot \left({m}_{\gamma }d{\xi }_{\gamma }+{m}_{3}d{x}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(g+{x}_{3}\kappa \right)\cdot \left({m}_{\gamma }d{\xi }_{\gamma }+{m}_{3}d{x}_{3}\right)\end{array}$

1. An infinitesimal line element dx in the undeformed shell can also be related to $d{\xi }_{\alpha },d{x}_{3}$ as

$dx=\frac{d\overline{r}}{d{\xi }_{\alpha }}d{\xi }_{\alpha }+{x}_{3}\frac{\partial {\overline{m}}_{3}}{\partial {\xi }_{\alpha }}d{\xi }_{\alpha }+d{x}_{3}{\overline{m}}_{3}$

This can be re-written as

$\begin{array}{l}dx=\left({\delta }_{\beta }^{\alpha }{\overline{m}}_{\alpha }\otimes {\overline{m}}^{\beta }+{\overline{m}}_{3}\otimes {\overline{m}}^{3}+{x}_{3}{\overline{\kappa }}_{\beta }^{\alpha }{\overline{m}}_{\alpha }\otimes {\overline{m}}^{\beta }\right)\cdot \left({\overline{m}}_{\gamma }d{\xi }_{\gamma }+{\overline{m}}_{3}d{x}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(\overline{g}+{x}_{3}\overline{\kappa }\right)\left({\overline{m}}_{\gamma }d{\xi }_{\gamma }+{\overline{m}}_{3}d{x}_{3}\right)\end{array}$

so that $\left({\overline{m}}_{\gamma }d{\xi }_{\gamma }+{\overline{m}}_{3}d{x}_{3}\right)={\left(\overline{g}+{x}_{3}\overline{\kappa }\right)}^{-1}dx$

1. Finally, note that

$\left({m}_{\beta }d{\xi }_{\beta }+{m}_{3}d{x}_{3}\right)=\left({m}_{i}\otimes {\overline{m}}^{i}\right)\cdot \left({\overline{m}}_{\gamma }d{\xi }_{\gamma }+{\overline{m}}_{3}d{x}_{3}\right)=\left({m}_{i}\otimes {\overline{m}}^{i}\right){\left(\overline{g}+{x}_{3}\overline{\kappa }\right)}^{-1}dx$

We can substitute this result into (2) above to see that

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}dy=\left(g+{x}_{3}\kappa \right)\left({m}_{i}\otimes {\overline{m}}^{i}\right){\left(\overline{g}+{x}_{3}\overline{\kappa }\right)}^{-1}dx$

and the deformation gradient can be read off as the coefficient of dx.

1. The approximate expression for F is obtained by assuming that for a thin shell

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(\overline{g}+{x}_{3}\overline{\kappa }\right)}^{-1}\approx \left(\overline{g}-{x}_{3}\overline{\kappa }\right)$

To see this, multiply out $\text{\hspace{0.17em}}\left(\overline{g}+{x}_{3}\overline{\kappa }\right)\left(\overline{g}-{x}_{3}\overline{\kappa }\right)$, recall that $\overline{g}\overline{g}=\overline{g},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{g}\overline{\kappa }=\overline{\kappa }\overline{g}=\overline{\kappa }$ and neglect the term of order ${x}_{3}^{2}$. Finally, substitute this approximation into the formula for F, multiply out the terms and neglect ${x}_{3}^{2}$ terms to obtain the approximation for F.

10.5.6 Other deformation measures.

It is straightforward to calculate any other deformation of interest from the deformation gradient.  A few examples that will be used in calculations to follow are listed below.

The inverse of the deformation gradient can be approximated by

$\begin{array}{l}{F}^{-1}=\left(\overline{g}+{x}_{3}\overline{\kappa }\right){\left({m}_{\alpha }\otimes {\overline{m}}^{\alpha }\right)}^{-1}{\left(}^{g}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx {\overline{m}}_{\alpha }\otimes {m}^{\alpha }-{x}_{3}\left({\kappa }_{\alpha }^{\beta }-{\overline{\kappa }}_{\alpha }^{\beta }\right){\overline{m}}_{\alpha }\otimes {m}^{\beta }+{\overline{m}}_{3}\otimes {m}^{3}\end{array}$

The velocity gradient tensor $L$, which relates the relative velocity $d\stackrel{˙}{y}$ of two material particles at positions $y$ and $y+dy$ in the deformed shell as $d\stackrel{˙}{y}=Ldy$ can be approximated by

$\begin{array}{l}L=\left({\stackrel{˙}{m}}_{i}\otimes {m}^{i}+{x}_{3}{\stackrel{˙}{\kappa }}_{\alpha \beta }{m}^{\alpha }\otimes {m}^{\beta }\right){\left({m}_{i}\otimes {m}^{i}+{x}_{3}{\kappa }_{\beta }^{\alpha }{m}_{\alpha }\otimes {m}^{\beta }\right)}^{-1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx \left({\stackrel{˙}{m}}_{i}\otimes {m}^{i}+{x}_{3}{\stackrel{˙}{\kappa }}_{\alpha \beta }{m}^{\alpha }\otimes {m}^{\beta }\right)\left({m}_{i}\otimes {m}^{i}-{x}_{3}{\kappa }_{\beta }^{\alpha }{m}_{\alpha }\otimes {m}^{\beta }\right)\end{array}$

where ${\stackrel{˙}{\kappa }}_{\alpha \beta }$ are the covariant components of the time derivative of the surface curvature tensor.

The Lagrange strain tensor can be approximated by

$E=\left({F}^{T}F-I\right)/2\approx \frac{1}{2}\left({g}_{\alpha \beta }-{\overline{g}}_{\alpha \beta }\right){\overline{m}}^{\alpha }\otimes {\overline{m}}^{\beta }+{x}_{3}\left({\kappa }_{\beta }^{\alpha }-{\overline{\kappa }}_{\beta }^{\alpha }\right){g}_{\lambda \alpha }{\overline{m}}^{\lambda }\otimes {\overline{m}}^{\beta }$

where terms of order ${x}_{3}^{2}$ have been neglected, we have used ${g}_{\alpha \beta }={m}_{\alpha }\cdot {m}_{\beta }$ and $I\equiv {\overline{g}}_{\alpha \beta }{\overline{m}}^{\alpha }\otimes {\overline{m}}^{\beta }$.

When we write constitutive equations relating forces to deformations, it is convenient to introduce two new strain measures defined as follows:

1. The mid-plane Lagrange strain tensor’ $\gamma ={\gamma }_{\alpha \beta }{\overline{m}}^{\alpha }\otimes {\overline{m}}^{\beta }=\frac{1}{2}\left({g}_{\alpha \beta }-{\overline{g}}_{\alpha \beta }\right){\overline{m}}^{\alpha }\otimes {\overline{m}}^{\beta }$,

where ${g}_{\alpha \beta }={m}_{\alpha }\cdot {m}_{\beta }$ and ${\overline{g}}_{\alpha \beta }={\overline{m}}_{\alpha }\cdot {\overline{m}}_{\beta }$.  The tensor quantifies length changes of infinitesimal material elements in the mid-plane of the shell, in the sense that the lengths $d\overline{s}$ and $ds$ of a line element before and after deformation are related by

$d{\xi }_{\alpha }{\overline{m}}_{\alpha }\cdot \gamma \cdot {\overline{m}}_{\beta }d{\xi }_{\beta }=\left({g}_{\alpha \beta }d{\xi }_{\alpha }d{\xi }_{\beta }-{\overline{g}}_{\alpha \beta }d{\xi }_{\alpha }d{\xi }_{\beta }\right)/2=\left(d{s}^{2}-d{\overline{s}}^{2}\right)/2$

1. The Curvature change tensor’ $\Delta \kappa =\Delta {\kappa }_{\lambda \beta }{\overline{m}}^{\lambda }\otimes {\overline{m}}^{\beta }=\left({\kappa }_{\beta }^{\alpha }-{\overline{\kappa }}_{\beta }^{\alpha }\right){g}_{\lambda \alpha }{\overline{m}}^{\lambda }\otimes {\overline{m}}^{\beta }$, which quantifies the additional stretch induced by bending and twisting the shell.

10.5.7 Representation of forces and moments in shells

The figure shows a generic cross-section of the shell, in the deformed configuration. To define measures of internal and external forces acting on the shell, we define the following variables

A basis $\left\{{m}_{1},{m}_{2},{m}_{3}\right\}$ with vectors chosen following the scheme described in 10.5.1.  Vector and tensor quantities will be quantified by their contravariant components in this basis

The body force acting on the rod $b$, or in component form $b={b}^{i}{m}_{i}$

The tractions acting on the exterior surface of the shell $t={t}^{i}{m}_{i}$.  It is convenient to define separate variables to characterize the tractions acting on the various parts of the shell, as indicated in the sketch: the upper surface of the shell (denoted by ${S}_{+}$ ) is subjected to traction ${t}_{+}^{i}$; the lower surface ${S}_{-}$ is subjected to ${t}_{-}^{i}$, while the surface around the edge of the shell ${S}_{e}$ is subjected to ${t}_{e}^{i}$

The Cauchy stress within the shell $\sigma ={\sigma }^{ij}{m}_{i}\otimes {m}_{j}$.

External forces and moments acting on the shell are characterized by

1. The external force per unit area acting on the shell, $p={p}^{i}{m}_{i}$.  The force components can be calculated from the tractions and body force acting on the shell as

${p}^{i}\left({\xi }_{\alpha }\right)={t}_{+}^{i}\left({\xi }_{\alpha }\right)+{t}_{-}^{i}\left({\xi }_{\alpha }\right)+h{\kappa }_{\alpha }^{\alpha }\left({t}_{+}^{i}\left({\xi }_{\alpha }\right)-{t}_{-}^{i}\left({\xi }_{\alpha }\right)\right)/2+\underset{-h/2}{\overset{h/2}{\int }}{b}^{i}\left({\xi }_{\alpha },{x}_{3}\right)\left(1+{x}_{3}{\kappa }_{\alpha }^{\alpha }\right)d{x}_{3}$

1. The external moment per unit area $q$ acting on the shell.  It is most convenient to express the external moment as  $q={q}^{\alpha }{m}_{3}×{m}_{\alpha }$ where the components ${q}^{\alpha }$ can be calculated from the tractions and body force as

${q}^{\alpha }\left({\xi }_{\beta }\right)=\left[{t}_{+}^{\alpha }\left({\xi }_{\beta }\right)-{t}_{-}^{\alpha }\left({\xi }_{\beta }\right)\right]h/2+\underset{-h/2}{\overset{h/2}{\int }}{x}_{3}{b}^{\alpha }\left({\xi }_{\beta },{x}_{3}\right)d{x}_{3}$

The vector q is work conjugate to the angular velocity $\omega ={\stackrel{˙}{\mu }}_{\alpha }{m}_{3}×{m}^{\alpha }$ of the normal to the mid-plane of the shell, in the sense that $q\cdot \omega ={q}^{\alpha }\left({m}_{3}×{m}_{\alpha }\right)\cdot {\stackrel{˙}{\mu }}_{\beta }\left({m}_{3}×{m}^{\beta }\right)={q}^{\alpha }{\stackrel{˙}{\mu }}_{\alpha }$ is the rate of work done by the external couple per unit area of the shell’s mid-plane.

1. The resultant force per unit length acting on the external edge of the shell.  The force per unit length can be expressed as components as $P={P}^{i}{m}_{i}$.  The components are related to the tractions acting on the external surface at the edge of the shell by ${P}^{i}=\underset{-h/2}{\overset{h/2}{\int }}{t}^{i}\left(1+{x}_{3}{\tau }^{\alpha }{\tau }^{\beta }{\kappa }_{\alpha \beta }\right)d{x}_{3}$
2. The resultant moment per unit length acting on the external edge of the shell.  The moment per unit length can be expressed as components as $Q={Q}^{\alpha }{m}_{3}×{m}_{\alpha }$.  The components are related to the tractions acting on external surface at the edge of the shell

${Q}^{\alpha }=\underset{-h/2}{\overset{h/2}{\int }}{t}^{\alpha }{x}_{3}\left(1+{x}_{3}{\tau }^{\alpha }{\tau }^{\beta }{\kappa }_{\alpha \beta }\right)d{x}_{3}$

Internal forces and moments within the shell are characterized by three surface tensors $T$, $V$ and $M$,  To visualize their physical significance, suppose that the shell is cut through so as to expose an internal surface, which lies perpendicular to the mid-plane of the shell.   Let $n={n}^{\alpha }{m}_{\alpha }$ denote a unit vector normal to the internal surface, and let ds denote an infinitesimal line element that lies in the both the exposed surface and the mid-plane of the shell.   The exposed surface is subjected to a distribution of traction, so that an small element of area with dimensions $ds×h$ is subjected to a resultant force $df=d{f}^{i}{m}_{i}$ and resultant moment $d\eta =d{\eta }^{\alpha }{m}_{\alpha }$.  These forces and moments are related to $T$, $V$ and $M$ as outlined below:

1.      The in-plane stress resultant tensor $T={T}^{\alpha \beta }{m}_{\alpha }\otimes {m}_{\beta }$ quantifies internal forces that tend to stretch and shear the shell in its own plane.  It is related to the internal tractions by $d{f}^{\alpha }{m}_{\alpha }=dsn\cdot T$, and its components can be calculated from the stress distribution in the shell as

${T}^{\alpha \beta }=\underset{-h/2}{\overset{h/2}{\int }}\left({\sigma }^{\alpha \beta }-{x}_{3}{\sigma }^{\gamma \alpha }{\kappa }_{\gamma }^{\beta }\right)\left(1+{x}_{3}{\kappa }_{\lambda }^{\lambda }\right)d{x}_{3}$

2.      The transverse shearing stress tensor $V={V}^{\beta }{m}_{\beta }\otimes {m}_{3}$ quantifies internal forces that act to impose the constraint that material fibers that are perpendicular to the mid-plane of the shell before deformation remain perpendicular to the mid-plane after deformation.  Strictly speaking, in Kirchoff shell theory it is a Lagrange multiplier, but can be regarded as quantifying the transverse shear force $d{f}^{3}{m}_{3}=dsn\cdot V$.  Its value cannot be computed from the deformation of the shell, because the transverse shearing has been neglected: instead, it must be determined by solving the equilibrium equations given in the next section.

3.      The internal moment tensor $M={M}^{\alpha \beta }{m}_{\alpha }\otimes \left({m}_{3}×{m}_{\beta }\right)$ characterizes internal bending and twisting moments in the shell.   It is related to the moment acting on internal through-thickness sections of the shell by $d\eta ={\eta }^{\alpha }{m}_{\alpha }=dsn\cdot M$.  The components ${M}^{\alpha \beta }$ can be calculated from the internal stresses in the shell as

${M}^{\alpha \beta }=\underset{-h/2}{\overset{h/2}{\int }}{x}_{3}\left({\sigma }^{\alpha \beta }-{x}_{3}{\sigma }^{\gamma \alpha }{\kappa }_{\gamma }^{\beta }\right)\left(1+{x}_{3}{\kappa }_{\lambda }^{\lambda }\right)d{x}_{3}$

The tensor $M$ is work conjugate to the gradient of the angular velocity of the normal to the mid-plane of the shell $\omega ={\stackrel{˙}{\mu }}_{\alpha }{m}_{3}×{m}^{\alpha }$, or alternatively, to the rate of change of curvature in the sense that

$\begin{array}{l}M:\left({m}^{\alpha }\otimes \frac{\partial \omega }{\partial {\xi }_{\alpha }}\right)=\left\{{M}^{\alpha \beta }{m}_{\alpha }\otimes \left({m}_{3}×{m}_{\beta }\right)\right\}:\left\{{m}^{\gamma }\otimes \frac{\partial }{\partial {\xi }_{\gamma }}\left\{{\stackrel{˙}{\mu }}_{\lambda }\left({m}_{3}×{m}^{\lambda }\right)\right\}\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={M}^{\alpha \beta }\left(\frac{\partial {\stackrel{˙}{\mu }}_{\beta }}{\partial {\xi }_{\alpha }}-{\stackrel{˙}{\mu }}_{\lambda }{\Gamma }_{\alpha \beta }^{\lambda }\right)={M}^{\alpha \beta }{\stackrel{˙}{\kappa }}_{\alpha \beta }\end{array}$

is the rate of work done by M per unit area of the mid-plane of the shell.

10.5.8 Equations of motion and boundary conditions

We consider a shell with thickness h and mass density $\rho$. The internal forces and moments must satisfy

${m}^{\alpha }\cdot \frac{\partial }{\partial {\xi }_{\alpha }}\left(V+T\right)+p=\rho ha\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}^{\alpha }\cdot \frac{\partial M}{\partial {\xi }_{\alpha }}+{m}_{\alpha }×\left[{m}^{\alpha }\cdot \left(V-T\right)\right]+q=\frac{\rho {h}^{3}}{12}\alpha ×{m}_{3}\text{\hspace{0.17em}}$

The operator ${m}^{\alpha }\cdot \left(\partial /\partial {\xi }_{\alpha }\right)$ represents the surface divergence, T, V and M are the internal forces defined in Sect 10.5.7; p and q are the external force and couple per unit area acting on the shell, a is the acceleration of the mid-plane and $\alpha$ is the angular acceleration of the unit vector normal to the mid-plane of the shell.  The two equations can be interpreted as linear and angular momentum balance for an infinitesimal element of the shell.  Note that:

1.      If the system is in static equilibrium, the right hand sides of all the equations of motion are zero.

2.      In addition, in many dynamic problems, the right hand sides of the angular momentum balance equations may be taken to be approximately zero.  For example, the rotational inertia may be ignored when modeling the vibration of a shell.  The rotational inertia terms can be important if the shell is rotating rapidly: for example, they would influence the out-of-plane vibration of a spinning disk.

The equations of motion can also be expressed in terms of components of the various force and moment tensors by substituting $T={T}^{\alpha \beta }{m}_{\alpha }\otimes {m}_{\beta }$, $V={V}^{\beta }{m}_{\beta }\otimes {m}_{3}$ $M={M}^{\alpha \beta }{m}_{\alpha }\otimes \left({m}_{3}×{m}_{\beta }\right)$, $p={p}^{i}{m}_{i}$, $q={q}^{\alpha }{m}_{3}×{m}_{\alpha }$, $a={a}^{i}{m}_{i}$ and $\alpha ={\stackrel{¨}{\mu }}^{\beta }{m}_{3}×{m}_{\beta }+{\stackrel{˙}{\mu }}_{\beta }{\stackrel{˙}{\mu }}^{\beta }{m}_{3}$ and recalling that

$\frac{\partial {m}_{\alpha }}{\partial {\xi }_{\gamma }}={\Gamma }_{\alpha \gamma }^{\lambda }{m}_{\lambda }-{\kappa }_{\alpha \gamma }{m}_{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {m}_{3}}{\partial {\xi }_{\gamma }}={\kappa }_{\gamma }^{\lambda }{m}_{\lambda }$

The result is

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

The last equation shows that the stress resultant and moment tensors are not symmetric.  The asymmetry is small, and is ignored in simplified versions of shell theory.  However, there are a few special shell geometries (a cylindrical shell subjected to torsional loading is one example) where neglecting the asymmetry can lead to substantial errors.

Edge boundary conditions for a shell are complicated and confusing.  To understand them, it is helpful to visualize the possible types of motion that can occur at the edge of a shell.  The edge of the shell is characterized by a curve C that lies in the mid-plane of the shell, encircling ${m}_{3}$ in a counterclockwise sense.   We let $s$ denote arc-length measured around C from some convenient origin, and use $\tau ={\tau }^{\alpha }{m}_{\alpha }$ and $n={m}_{3}×\tau$ denote unit vectors tangent and normal to C.  Elements of the shell that lie on C have four independent degrees of freedom, as follows:

1. The material element can have an arbitrary velocity, with three components $\delta v=\delta {v}_{i}{m}^{i}$
2. The material element can rotate about the tangent vector $\tau$. To visualize this motion, imagine that the shell is supported around C by a hinge.

The motion of the edge of the shell can be prescribed by constraining one or more of these degrees of freedom.   Alternatively, the edge of the shell can be subjected to one or more of four generalized forces, which are work-conjugate to these degrees of freedom.  The generalized forces can be expressed in terms of the forces $P={P}^{i}{m}_{i}$ and couples $Q={Q}^{\alpha }{m}_{3}×{m}_{\alpha }$ acting on the edge of the shell as

1. ${P}^{\beta }+{Q}^{\lambda }{\kappa }_{\lambda }^{\beta }$ is work-conjugate to the in-plane displacement of the shell $\delta {v}_{\beta }$
2. ${P}^{3}-\frac{\partial }{\partial s}\left[{Q}^{\beta }{\tau }_{\beta }\right]$ is work-conjugate to the out-of-plane displacement of the edge of the shell $\delta {v}_{3}$
3. ${Q}^{\beta }{n}_{\beta }$ is work-conjugate to the rotation of the shell about its edge.

The four boundary conditions can be expressed in terms of these forces as follows:

$\begin{array}{l}{n}_{\alpha }{T}^{\alpha \beta }+{n}_{\alpha }{M}^{\alpha \lambda }{\kappa }_{\lambda }^{\beta }={P}^{\beta }+{Q}^{\lambda }{\kappa }_{\lambda }^{\beta }\\ {n}_{\alpha }{V}^{\alpha }+\frac{\partial }{\partial s}\left[{n}_{\alpha }{M}^{\alpha \beta }{\tau }_{\beta }\right]={P}^{3}+\frac{\partial }{\partial s}\left[{Q}^{\beta }{\tau }_{\beta }\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ {n}_{\alpha }{M}^{\alpha \beta }{n}_{\beta }={Q}^{\beta }{n}_{\beta }\end{array}$

Derivation: Measures of internal force, the equilibrium equations, and the boundary conditions emerge naturally from the principle of virtual work.  The principle of virtual work states that, for any deformable solid that is in static equilibrium, the Cauchy stress distribution must satisfy

$\underset{V}{\int }\sigma :\delta LdV+\underset{V}{\int }\rho a\cdot \delta \stackrel{˙}{y}dV-\underset{V}{\int }b\cdot \delta \stackrel{˙}{y}dV-\underset{S}{\int }t\cdot \delta \stackrel{˙}{y}dA=0$

for all virtual velocity fields $\delta \stackrel{˙}{y}$ and compatible velocity gradients $\delta L$.  The virtual velocity field in the shell must have the same general form as the actual velocity, as outlined in Section 10.2.4.  The virtual velocity can therefore be characterized by the virtual velocity of the mid-plane of the shell $\delta v=\delta {v}_{i}{m}^{i}$.  It is convenient to introduce the time derivative of the normal vector to the plate’s mid-plane $\delta {\stackrel{˙}{m}}_{3}=\delta {\stackrel{˙}{\mu }}_{\alpha }{m}^{\alpha }$ as an additional kinematic variable, which must of course be compatible with $\delta v$.  We will show the following:

The virtual work principle can be expressed in terms of the generalized deformation measures and forces defined in the preceding sections as

$\begin{array}{l}\underset{A}{\int }{T}^{\alpha \beta }\left\{\frac{\partial \delta {v}_{\beta }}{\partial {\xi }_{\alpha }}-\delta {v}_{\lambda }{\Gamma }_{\beta \alpha }^{\lambda }+\delta {v}_{3}{\kappa }_{\beta \alpha }\right\}dA+\underset{A}{\int }{M}^{\alpha \beta }\left(\frac{\partial \delta {\stackrel{˙}{\mu }}_{\alpha }}{\partial {\xi }_{\beta }}-\delta {\stackrel{˙}{\mu }}_{\lambda }{\Gamma }_{\alpha \beta }^{\lambda }\right)dA\\ \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{A}{\int }h\rho {a}^{i}\delta {v}_{i}dA+\underset{A}{\int }\frac{\rho {h}^{3}}{12}{\stackrel{¨}{\mu }}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }dA\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{A}{\int }{p}^{i}\delta {v}_{i}dA-\underset{A}{\int }{q}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }dA-\underset{C}{\int }{P}^{i}\delta {v}_{i}ds-\underset{C}{\int }{Q}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }ds=0\end{array}$

If the virtual work equation is satisfied for all $\delta {v}_{i}$ and compatible $\delta {\stackrel{˙}{\mu }}_{\alpha }$, then the internal forces and moments must satisfy the following equilibrium equations

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

as well as the boundary conditions listed above.

The last equilibrium equation ${T}^{12}-{T}^{21}+{M}^{\alpha 1}{\kappa }_{\alpha }^{2}-{M}^{\alpha 2}{\kappa }_{\alpha }^{1}=0$ does not emerge from the virtual work principle. Instead, this equation is a consequence of the symmety of the Cauchy stress tensor ${\sigma }^{\alpha \beta }$, as shown below. It is automatically satisfied if the components  ${T}^{\alpha \beta }$ and ${M}^{\alpha \beta }$ are calculated by integrating the stresses through the thickness of the shell.  However,  for some statically determinate shell problems it is possible to avoid evaluating these integrals explicitly, in which case the equilibrium equation is useful.

Expressing the virtual work equation in terms of generalized force measures is a straightforward but lengthy algebraic exercise.

1. When applying the virtual work principle, we will need to integrate over the volume of the shell.  It is convenient to write the volume integral as separate integrals over the mid-plane of the shell and through its thickness, as follows

$\underset{V}{\int }dV=\underset{A}{\iint }\underset{-h/2}{\overset{h/2}{\int }}\left(\frac{\partial y}{\partial {\xi }_{1}}×\frac{\partial y}{\partial {\xi }_{2}}\right)\cdot \frac{\partial y}{\partial {x}_{3}}d{x}_{3}d{\xi }_{1}d{\xi }_{2}$

Recall that $y=r+{x}_{3}{m}_{3}$, so that

$\frac{\partial y}{\partial {\xi }_{\alpha }}={m}_{\alpha }+{x}_{3}{\kappa }_{\alpha }^{\beta }{m}_{\beta }$

Therefore $\left(\frac{\partial y}{\partial {\xi }_{1}}×\frac{\partial y}{\partial {\xi }_{2}}\right)\cdot \frac{\partial y}{\partial {x}_{3}}\approx \left(1+{x}_{3}{\kappa }_{\alpha }^{\alpha }\right)\left({m}_{1}×{m}_{2}\right)\cdot {m}_{3}$, where the term of order ${x}_{3}^{2}$ has been neglected.  Substituting this result into the volume integral yields

$\underset{V}{\int }dV\approx \underset{A}{\iint }\underset{-h/2}{\overset{h/2}{\int }}\left(1+{x}_{3}{\kappa }_{\alpha }^{\alpha }\right)\left({m}_{1}×{m}_{2}\right)\cdot {m}_{3}d{\xi }_{1}d{\xi }_{2}d{x}_{3}\equiv \underset{A}{\int }\underset{-h/2}{\overset{h/2}{\int }}\left(1+{x}_{3}{\kappa }_{\alpha }^{\alpha }\right)d{x}_{3}dA$

where the area integral is understood to be taken over the mid-plane of the shell.

1. Similarly, the integrals over the outer surface of the shell can be separated into integrals taken over the upper and lower surfaces of the shell ( ${S}_{+}$ and ${S}_{-}$ ), together with an integral over the surface at the edge of the shell ${S}_{e}$, as follows

$\underset{S}{\int }dA=\underset{S+}{\int }dA+\underset{S-}{\int }dA+\underset{{S}_{e}}{\int }dA$

Following the procedure in step (1), the integrals over ${S}_{+}$ and ${S}_{-}$ can be expressed in terms of integrals taken over the mid-plane of the shell as

$\underset{S+}{\int }dA=\underset{A}{\int }\left(1+h{\kappa }_{\alpha }^{\alpha }/2\right)dA\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{S-}{\int }dA=\underset{A}{\int }\left(1-h{\kappa }_{\alpha }^{\alpha }/2\right)dA$

The integral over ${S}_{e}$ can be reduced to a line integral taken around the curve(s) bounding the edge of the shell, as

$\underset{{S}_{e}}{\int }dA=\underset{C}{\int }\underset{-h/2}{\overset{h/2}{\int }}\left(1+{x}_{3}{\tau }^{\alpha }{\tau }^{\beta }{\kappa }_{\alpha \beta }\right)d{x}_{3}ds$

The procedure to derive this result is very similar to the steps required to simplify the volume integral and is left as an exercise.

1. Next, consider the integrand

$\begin{array}{l}\sigma :\delta L\approx \sigma :\left[\left(\delta {\stackrel{˙}{m}}_{i}\otimes {m}^{i}+{x}_{3}\delta {\stackrel{˙}{\kappa }}_{\alpha \beta }{m}^{\alpha }\otimes {m}^{\beta }\right)\left({m}_{i}\otimes {m}^{i}-{x}_{3}{\kappa }_{\beta }^{\alpha }{m}_{\alpha }\otimes {m}^{\beta }\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[\sigma {\left({m}_{i}\otimes {m}^{i}-{x}_{3}{\kappa }_{\beta }^{\alpha }{m}_{\alpha }\otimes {m}^{\beta }\right)}^{T}\right]:\left(\delta {\stackrel{˙}{m}}_{i}\otimes {m}^{i}+{x}_{3}\delta {\stackrel{˙}{\kappa }}_{\alpha \beta }{m}^{\alpha }\otimes {m}^{\beta }\right)\end{array}$

To reduce this to a scalar combination of the components of the various tensors and vectors, substitute $\sigma ={\sigma }^{ij}{m}_{i}\otimes {m}_{j}$, together with the kinematic formulas:

$\begin{array}{l}\delta {\stackrel{˙}{m}}_{\alpha }=\frac{\partial \delta {v}_{\beta }}{\partial {\xi }_{\alpha }}{m}^{\beta }-d{v}_{\beta }{\Gamma }_{\lambda \alpha }^{\beta }{m}^{\lambda }+\delta {v}_{3}{\kappa }_{\beta \alpha }{m}^{\beta }+\left(\frac{\partial \delta {v}_{3}}{\partial {\xi }_{\alpha }}-\delta {v}_{\beta }{\kappa }_{\alpha }^{\beta }\right){m}^{3}\\ \delta {\stackrel{˙}{m}}_{3}=-\left(\frac{\partial {v}_{3}}{\partial {\xi }_{\alpha }}-{v}_{\beta }{\kappa }_{\alpha }^{\beta }\right){m}^{\alpha }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\delta {\stackrel{˙}{\kappa }}_{\alpha \beta }{m}^{\alpha }=\left(\frac{\partial \delta {\stackrel{˙}{\mu }}_{\alpha }}{\partial {\xi }_{\beta }}{m}^{\alpha }-\delta {\stackrel{˙}{\mu }}_{\lambda }{\Gamma }_{\alpha \beta }^{\lambda }{m}^{\alpha }\right)\end{array}$

with the result

$\begin{array}{l}\sigma :\delta L\approx \left({\sigma }^{\alpha \beta }-{x}_{3}{\sigma }^{\gamma \alpha }{\kappa }_{\gamma }^{\beta }\right)\left\{\frac{\partial \delta {v}_{\beta }}{\partial {\xi }_{\alpha }}-\delta {v}_{\lambda }{\Gamma }_{\beta \alpha }^{\lambda }+\delta {v}_{3}{\kappa }_{\beta \alpha }\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}_{3}\left({\sigma }^{\alpha \beta }-{x}_{3}{\sigma }^{\gamma \alpha }{\kappa }_{\gamma }^{\beta }\right)\left(\frac{\partial {\stackrel{˙}{\mu }}_{\alpha }}{\partial {\xi }_{\beta }}-{\stackrel{˙}{\mu }}_{\lambda }{\Gamma }_{\alpha \beta }^{\lambda }\right)\end{array}$

1. Substituting this result into the first integral of the virtual work principle, reducing the volume integral to an integral over the mid-plane of the shell, and using the definitions of ${T}^{\alpha \beta }$ and ${M}^{\alpha \beta }$ gives

$\underset{V}{\int }\sigma :\delta LdV=\underset{A}{\int }{T}^{\alpha \beta }\left\{\frac{\partial \delta {v}_{\beta }}{\partial {\xi }_{\alpha }}-\delta {v}_{\lambda }{\Gamma }_{\beta \alpha }^{\lambda }+\delta {v}_{3}{\kappa }_{\beta \alpha }\right\}dA+\underset{A}{\int }{M}^{\alpha \beta }\left(\frac{\partial {\stackrel{˙}{\mu }}_{\alpha }}{\partial {\xi }_{\beta }}-{\stackrel{˙}{\mu }}_{\lambda }{\Gamma }_{\alpha \beta }^{\lambda }\right)dA$

1. Similar manipulations can be used to reduce the remaining terms in the virtual work principle to

$\underset{V}{\int }\rho a\cdot \delta \stackrel{˙}{y}dV=\underset{A}{\int }h\rho {a}^{i}\delta {v}_{i}dA+\underset{A}{\int }\frac{\rho {h}^{3}}{12}{\stackrel{¨}{\mu }}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }dA$

$\underset{V}{\int }b\cdot \delta \stackrel{˙}{y}dV+\underset{S}{\int }t\cdot \delta \stackrel{˙}{y}dA=\underset{A}{\int }{p}^{i}\delta {v}_{i}dA+\underset{A}{\int }{q}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }dA+\underset{C}{\int }{P}^{i}\delta {v}_{i}ds+\underset{C}{\int }{Q}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }ds$

Substituting the equations in (4) and (5) into the virtual work equation gives the first result.

Next, we show that the equilibrium equations and boundary conditions follow from the virtual work principle.

1. The virtual work equation must first be augmented by a Lagrange multiplier to enforce compatibility between the velocity field $\delta v=\delta {v}_{i}{m}^{i}$ and the time derivative of the vector normal to the shell’s mid plane $\delta {\stackrel{˙}{m}}_{3}=\delta {\stackrel{˙}{\mu }}_{\alpha }{m}^{\alpha }$.  To this end, we regard the unit vector ${m}_{3}$ as an independent degree of freedom, and introduce a vector valued Lagrange multiplier ${V}^{\alpha }{m}_{\alpha }$ which must satisfy

$\underset{A}{\int }\left({m}_{3}-\frac{{m}_{1}×{m}_{2}}{|{m}_{1}||{m}_{1}|}\right)\cdot \delta {V}^{\alpha }{m}_{\alpha }dA+\underset{A}{\int }\left\{\delta {\stackrel{˙}{\mu }}_{\alpha }{m}^{\alpha }+\left(\frac{\partial \delta {v}_{3}}{\partial {\xi }_{\alpha }}-\delta {v}_{\beta }{\kappa }_{\alpha }^{\beta }\right){m}^{\alpha }\right\}\cdot {V}^{\gamma }{m}_{\gamma }dA=0$

for all admissible variations $\delta {\stackrel{˙}{V}}^{\alpha }{m}_{\alpha }$, $\delta v=\delta {v}_{i}{m}^{i}$ and $\delta {\stackrel{˙}{m}}_{3}=\delta {\stackrel{˙}{\mu }}_{\alpha }{m}^{\alpha }$.  The second integral can simply be added to the virtual work equation to ensure compatibility of $\delta {\stackrel{˙}{m}}_{3}=\delta {\stackrel{˙}{\mu }}_{\alpha }{m}^{\alpha }$ and $\delta v=\delta {v}_{i}{m}^{i}$

1. The augmented virtual work equation now reads

$\begin{array}{l}\underset{A}{\int }{T}^{\alpha \beta }\left\{\frac{\partial \delta {v}_{\beta }}{\partial {\xi }_{\alpha }}-\delta {v}_{\lambda }{\Gamma }_{\beta \alpha }^{\lambda }+\delta {v}_{3}{\kappa }_{\beta \alpha }\right\}dA+\underset{A}{\int }{M}^{\alpha \beta }\left(\frac{\partial {\stackrel{˙}{\mu }}_{\alpha }}{\partial {\xi }_{\beta }}-{\stackrel{˙}{\mu }}_{\lambda }{\Gamma }_{\alpha \beta }^{\lambda }\right)dA+\underset{A}{\int }h\rho {a}^{i}\delta {v}_{i}dA+\underset{A}{\int }\frac{\rho {h}^{3}}{12}{\stackrel{¨}{\mu }}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }dA\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{A}{\int }\left\{\delta {\stackrel{˙}{\mu }}_{\alpha }+\left(\frac{\partial \delta {v}_{3}}{\partial {\xi }_{\alpha }}-\delta {v}_{\beta }{\kappa }_{\alpha }^{\beta }\right)\right\}{V}^{\alpha }dA\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{A}{\int }{p}^{i}\delta {v}_{i}dA-\underset{A}{\int }{q}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }dA-\underset{C}{\int }{P}^{i}\delta {v}_{i}ds-\underset{C}{\int }{Q}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }ds=0\end{array}$

1. The remaining steps are routine, but fiddly.  The partial derivatives of $\delta {v}_{\alpha }$ and $\delta {\stackrel{˙}{\mu }}_{\alpha }$ must be removed by integrating by parts.  This is accomplished by applying the surface divergence theorem, which states that if $w$ is a differentiable vector field on an area A of the surface, and C is the curve bounding A, then

$\underset{A}{\int }{m}^{\alpha }\cdot \frac{\partial w}{\partial {\xi }_{\alpha }}dA=\underset{C}{\int }n\cdot wds$

where $n$ is the outward normal to C.  To see how to use this theorem, consider

$\underset{A}{\int }{T}^{\alpha \beta }\left(\frac{\partial \delta {v}_{\beta }}{\partial {\xi }_{\alpha }}-\delta {v}_{\lambda }{\Gamma }_{\beta \alpha }^{\lambda }\right)dA\equiv \underset{A}{\int }\left({T}^{\alpha \beta }{m}_{\alpha }\otimes {m}_{\beta }\right):\left({m}^{\gamma }\otimes \frac{\partial \delta {v}_{\lambda }{m}^{\lambda }}{\partial {\xi }_{\gamma }}\right)dA$

The integrand can be re-written as

$T:\left({m}^{\gamma }\otimes \frac{\partial \delta v}{\partial {\xi }_{\gamma }}\right)={m}^{\gamma }\cdot \frac{\partial }{\partial {\xi }_{\gamma }}\left(T\delta v\right)-{m}^{\gamma }\cdot \frac{\partial T}{\partial {\xi }_{\gamma }}\delta v$

Applying the surface divergence theorem to the first term on the right hand side of this equation shows that

$\underset{A}{\int }T:\left({m}^{\gamma }\otimes \frac{\partial \delta v}{\partial {\xi }_{\gamma }}\right)dA=\underset{C}{\int }n\cdot T\delta vds-\underset{A}{\int }{m}^{\gamma }\cdot \frac{\partial T}{\partial {\xi }_{\gamma }}\delta vdA$

Finally, substituting $T={T}^{\alpha \beta }{m}_{\alpha }\otimes {m}_{\beta }$ and $\delta v=\delta {v}_{\alpha }{m}^{\alpha }$ and remembering to differentiate the basis vectors gives the component form

$\underset{A}{\int }{T}^{\alpha \beta }\left(\frac{\partial \delta {v}_{\alpha }}{\partial {\xi }_{\beta }}-\delta {v}_{\lambda }{\Gamma }_{\alpha \beta }^{\lambda }\right)dA=\underset{C}{\int }{n}_{\alpha }{T}^{\alpha \beta }\delta {v}_{\beta }ds-\underset{A}{\int }\left(\frac{\partial {T}^{\alpha \beta }}{\partial {\xi }_{\alpha }}+{T}^{\alpha \beta }{\Gamma }_{\alpha \gamma }^{\gamma }+{T}^{\alpha \gamma }{\Gamma }_{\gamma \alpha }^{\beta }\right)\delta {v}_{\beta }dA$

1. Applying the procedure outlined in step (3) to similar terms, the virtual work equation can be re-written as

$\begin{array}{l}-\underset{A}{\int }\left\{\frac{\partial {T}^{\alpha \beta }}{\partial {\xi }_{\alpha }}+{T}^{\alpha \beta }{\Gamma }_{\alpha \gamma }^{\gamma }+{T}^{\alpha \gamma }{\Gamma }_{\gamma \alpha }^{\beta }+{V}^{\alpha }{\kappa }_{\alpha }^{\beta }+{p}^{\beta }-\rho h{a}^{\beta }\right\}\delta {v}_{\beta }\\ -\underset{A}{\int }\left\{\frac{\partial {V}^{\alpha }}{\partial {\xi }_{\alpha }}+{V}^{\alpha }{\Gamma }_{\alpha \beta }^{\beta }-{T}^{\alpha \beta }{\kappa }_{\alpha \beta }+{p}^{3}-\rho h{a}^{3}\right\}\delta {v}_{3}\\ -\underset{A}{\int }\left\{\frac{\partial {M}^{\alpha \beta }}{\partial {\xi }_{\alpha }}+{M}^{\alpha \beta }{\Gamma }_{\alpha \gamma }^{\gamma }+{M}^{\alpha \gamma }{\Gamma }_{\gamma \alpha }^{\beta }-{V}^{\beta }+{q}^{\beta }-\frac{\rho {h}^{3}}{12}{\stackrel{¨}{\mu }}^{\beta }\right\}\delta {\stackrel{˙}{\mu }}_{\beta }\\ +\underset{C}{\int }{n}_{\alpha }{T}^{\alpha \beta }\delta {v}_{\beta }ds+\underset{C}{\int }{n}_{\alpha }{M}^{\alpha \beta }\delta {\stackrel{˙}{\mu }}_{\beta }ds+\underset{C}{\int }{n}_{\alpha }{V}^{\alpha }\delta {v}_{3}ds-\underset{C}{\int }{P}^{i}\delta {v}_{i}ds-\underset{C}{\int }{Q}^{\alpha }\delta {\stackrel{˙}{\mu }}_{\alpha }ds=0\end{array}$

This equation must be satisfied for all $\delta {v}_{i}$ and $\delta {\stackrel{˙}{\mu }}_{\alpha }$, which immediately gives the equilibrium equations.

1. Some further algebra is required to derive the boundary conditions. It is tempting to conclude that coefficients of $\delta {v}_{i}$ and $\delta {\stackrel{˙}{\mu }}_{\alpha }$ in the boundary terms must all vanish, but this is not the case, because $\delta {v}_{i}$ and $\delta {\stackrel{˙}{\mu }}_{\alpha }$ are related by compatibility equations. The boundary terms must be expressed in terms of four independent degrees of freedom.  To this end, recall that ${\stackrel{˙}{\mu }}_{\alpha }=-\left(\partial {v}_{3}/\partial {\xi }_{\alpha }-{v}_{\beta }{\kappa }_{\alpha }^{\beta }\right)$, so the integral around the boundary can be re-written as

$\begin{array}{l}\underset{C}{\int }\left\{{n}_{\alpha }{T}^{\alpha \beta }-{n}_{\alpha }{M}^{\alpha \lambda }{\kappa }_{\lambda }^{\beta }-{P}^{\beta }-{Q}^{\lambda }{\kappa }_{\lambda }^{\beta }\right\}\delta {v}_{\beta }ds+\underset{C}{\int }\left({n}_{\alpha }{V}^{\alpha }-{P}^{3}\right)\delta {v}_{3}ds\\ \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}}\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}}\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}}\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}}\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}}\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}}-\underset{C}{\int }\left({n}_{\alpha }{M}^{\alpha \beta }-{Q}^{\beta }\right)\frac{\partial \delta {v}_{3}}{\partial {\xi }_{\beta }}ds=0\end{array}$

1. The vector $\partial \delta {v}_{3}/\partial {\xi }_{\alpha }$ can be expressed in terms of components parallel and perpendicular to the boundary of the plate C, as

$\frac{\partial \delta {v}_{3}}{\partial {\xi }_{\alpha }}=\frac{\partial \delta {v}_{3}}{\partial s}{\tau }_{\alpha }+\delta \stackrel{˙}{\theta }{n}_{\alpha }$

Here, $\tau ={\tau }_{\alpha }{m}^{\alpha }$ and $n={n}_{\alpha }{m}^{\alpha }={m}_{3}×\tau$ represent unit vectors tangent and normal to C, and $\delta \stackrel{˙}{\theta }$ is an independent degree of freedom that represents the rotation of the shell about $\tau$.  Finally, we integrate by parts to see that

$\begin{array}{l}\underset{C}{\int }\left({n}_{\alpha }{M}^{\alpha \beta }-{Q}^{\beta }\right)\frac{\partial \delta {v}_{3}}{\partial {\xi }_{\beta }}ds=\underset{C}{\int }\left({n}_{\alpha }{M}^{\alpha \beta }-{Q}^{\beta }\right){\tau }_{\beta }\frac{\partial \delta {v}_{3}}{\partial s}ds+\underset{C}{\int }\left({n}_{\alpha }{M}^{\alpha \beta }-{Q}^{\beta }\right){n}_{\beta }\delta \stackrel{˙}{\theta }ds\\ \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}}\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}}\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}}\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}}=-\underset{C}{\int }\frac{\partial }{\partial s}\left\{\left({n}_{\alpha }{M}^{\alpha \beta }-{Q}^{\beta }\right){\tau }_{\beta }\right\}\delta {v}_{3}ds+\underset{C}{\int }\left({n}_{\alpha }{M}^{\alpha \beta }-{Q}^{\beta }\right){n}_{\beta }\delta \stackrel{˙}{\theta }ds\end{array}$

(The terms associated with the ends of C vanish because C is a closed curve).

1. Substituting the result of (6) back into (5) gives

$\begin{array}{l}\underset{C}{\int }\left\{{n}_{\alpha }{T}^{\alpha \beta }-{n}_{\alpha }{M}^{\alpha \lambda }{\kappa }_{\lambda }^{\beta }-{P}^{\beta }-{Q}^{\lambda }{\kappa }_{\lambda }^{\beta }\right\}\delta {v}_{\beta }ds\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{C}{\int }\left({n}_{\alpha }{V}^{\alpha }+\frac{\partial }{\partial s}\left[{n}_{\alpha }{M}^{\alpha \beta }{\tau }_{\beta }\right]-{P}^{3}-\frac{\partial }{\partial s}\left[{Q}^{\beta }{\tau }_{\beta }\right]\right)\delta {v}_{3}ds-\underset{C}{\int }\left({n}_{\alpha }{M}^{\alpha \beta }-{Q}^{\beta }\right){n}_{\beta }\delta \stackrel{˙}{\theta }ds=0\end{array}$

This condition must be satisfied for all $\delta {v}_{i},\delta \stackrel{˙}{\theta }$, which gives the boundary conditions.

Finally, we must derive the last equilibrium equation ${T}^{12}-{T}^{21}+{M}^{\alpha 1}{\kappa }_{\alpha }^{2}-{M}^{\alpha 2}{\kappa }_{\alpha }^{1}=0$.  Using the definitions of ${T}^{\alpha \beta },{M}^{\alpha \beta }$, and noting that ${\sigma }^{12}={\sigma }^{21}$, it is straightforward to show that

$\begin{array}{l}{T}^{12}-{T}^{21}=\underset{-h/2}{\overset{h/2}{\int }}{x}_{3}\left({\sigma }^{\gamma 2}{\kappa }_{\gamma }^{1}-{\sigma }^{\gamma 1}{\kappa }_{\gamma }^{2}\right)\left(1+{x}_{3}{\kappa }_{\lambda }^{\lambda }\right)d{x}_{3}\\ {M}^{\alpha 1}{\kappa }_{\alpha }^{2}-{M}^{\alpha 2}{\kappa }_{\alpha }^{1}=\underset{-h/2}{\overset{h/2}{\int }}{x}_{3}\left({\sigma }^{\gamma 1}{\kappa }_{\gamma }^{2}-{\sigma }^{\gamma 2}{\kappa }_{\gamma }^{1}\right)\left(1+{x}_{3}{\kappa }_{\lambda }^{\lambda }\right)d{x}_{3}\end{array}$

Adding these two equations gives the last equilibrium equation.

10.5.9 Constitutive equations relating forces to deformation measures in elastic shells

The internal forces in a shell are related to its deformation by the stress-strain law for the material.  Here, we give force-deformation equations for an isotropic elastic shell which experiences small shape changes (but possibly large rotations).

Shape changes are characterized using the following deformation measures, defined in Sections 10.5.6

1. The in-plane components of the metric tensors for the mid-plane of the shell before and after deformation are denoted ${\overline{g}}_{\alpha \beta }={\overline{m}}_{\alpha }\cdot {\overline{m}}_{\beta }$ ${g}_{\alpha \beta }={m}_{\alpha }\cdot {m}_{\beta }$
2. The mid-plane Lagrange strain tensor’ $\gamma ={\gamma }_{\alpha \beta }{\overline{m}}^{\alpha }\otimes {\overline{m}}^{\beta }=\frac{1}{2}\left({g}_{\alpha \beta }-{\overline{g}}_{\alpha \beta }\right){\overline{m}}^{\alpha }\otimes {\overline{m}}^{\beta }$,
3. The Curvature change tensor’ $\Delta \kappa =\Delta {\kappa }_{\lambda \beta }{\overline{m}}^{\lambda }\otimes {\overline{m}}^{\beta }=\left({\kappa }_{\beta }^{\alpha }-{\overline{\kappa }}_{\beta }^{\alpha }\right){g}_{\lambda \alpha }{\overline{m}}^{\lambda }\otimes {\overline{m}}^{\beta }$, which quantifies the bending and twisting the shell.

Internal forces are characterized using the stress resultant tensor components ${T}^{\alpha \beta }$ and internal moment components ${M}^{\alpha \beta }$ defined in Section 10.5.7.

The shell is assumed to have a uniform thickness h, and is assumed to be made from an isotropic, linear elastic solid, with Young’s modulus E and Poisson’s ratio $\nu$.  We assume for simplicity that the shell is homogeneous, and neglect thermal expansion (the effects of thermal expansion are included an example problem solved in Section 10.7.4)

It is convenient to introduce a plane stress elasticity tensor with components

${D}^{\alpha \beta \rho \mu }=\frac{E}{2\left(1-{\nu }^{2}\right)}\left(\left({\overline{g}}^{\alpha \rho }{\overline{g}}^{\beta \mu }+{\overline{g}}^{\alpha \mu }{\overline{g}}^{\beta \rho }\right)\left(1-\nu \right)+2\nu {\overline{g}}^{\alpha \beta }{\overline{g}}^{\rho \mu }\right)$

The force-deformation relations can then be expressed as

${T}^{\alpha \beta }=h{D}^{\alpha \beta \rho \lambda }{\gamma }_{\rho \lambda }+\frac{{h}^{3}}{12}\left({\kappa }_{\mu }^{\mu }{\delta }_{\theta }^{\beta }-{\kappa }_{\theta }^{\beta }\right){D}^{\theta \alpha \rho \lambda }\Delta {\kappa }_{\rho \lambda }-\frac{{h}^{3}}{12}{\kappa }_{\mu }^{\mu }{D}^{\theta \alpha \rho \lambda }{\kappa }_{\theta }^{\beta }{\gamma }_{\rho \lambda }$

${M}^{\alpha \beta }=\frac{{h}^{3}}{12}{D}^{\alpha \beta \rho \lambda }\Delta {\kappa }_{\rho \lambda }+\frac{{h}^{3}}{12}\left({\kappa }_{\mu }^{\mu }{\delta }_{\theta }^{\beta }-{\kappa }_{\theta }^{\beta }\right){D}^{\theta \alpha \rho \lambda }{\gamma }_{\rho \lambda }$

For all but a few very rare shell geometries these expressions may be approximated by

${T}^{\alpha \beta }\approx h{D}^{\alpha \beta \rho \lambda }{\gamma }_{\rho \lambda }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{M}^{\alpha \beta }\approx \text{\hspace{0.17em}}\frac{{h}^{3}}{12}{D}^{\alpha \beta \rho \lambda }\Delta {\kappa }_{\rho \lambda }$

Derivation

1. We have assumed that the material in the shell experiences small distortions, but arbitrary rotations.  Material behavior can therefore be modeled using the generalized Hooke’s law described in Section 3.3, which relates the Material stress to the Lagrange strain using the isotropic linear elastic constitutive equations.
2. We assume that the shell is in a state of plane stress, so that the material stress tensor has the form  $\Sigma ={\Sigma }^{\alpha \beta }{\overline{m}}_{\alpha }\otimes {\overline{m}}_{\beta }$
3. The Lagrange strain  tensor is given in Section 10.5.6 as

$E\approx {\gamma }_{\alpha \beta }{\overline{m}}^{\alpha }\otimes {\overline{m}}^{\beta }+{x}_{3}\left({\kappa }_{\beta }^{\alpha }-{\overline{\kappa }}_{\beta }^{\alpha }\right){g}_{\lambda \alpha }{\overline{m}}^{\lambda }\otimes {\overline{m}}^{\beta }$

1. The material stress is related to the Lagrange strain by the plane stress version of the linear elastic constitutive equations, which can be expressed as

$\Sigma =\frac{E}{\left(1+\nu \right)}\left\{E+\frac{\nu }{1-\nu }\left(\overline{g}:E\right)\overline{g}\right\}$

where $\overline{g}={\overline{g}}^{\alpha \beta }{\overline{m}}_{\alpha }\otimes {\overline{m}}_{\beta }$  is the in-plane component of the metric tensor associated with the undeformed shell (this replaces the identity tensor in the Cartesian version of the constitutive equations)

1. The Cauchy stress is related to the material stress by $\sigma =F\Sigma {F}^{T}/J\approx F\Sigma {F}^{T}$.  Substituting the formulas for $\Sigma$ from (4) and approximating F as $F\approx {m}_{i}\otimes {\overline{m}}^{i}$, we find after some algebra that

$\sigma ={\sigma }^{\alpha \beta }{m}_{\alpha }\otimes {m}_{\beta }\approx {D}^{\alpha \beta \lambda \rho }\left({\gamma }_{\lambda \rho }+{x}_{3}\left({\kappa }_{\rho }^{\mu }-{\overline{\kappa }}_{\rho }^{\mu }\right){g}_{\lambda \mu }\right){m}_{\alpha }\otimes {m}_{\beta }$

where

${D}^{\alpha \beta \rho \mu }=\frac{E}{2\left(1-{\nu }^{2}\right)}\left(\left({\overline{g}}^{\alpha \rho }{\overline{g}}^{\beta \mu }+{\overline{g}}^{\alpha \mu }{\overline{g}}^{\beta \rho }\right)\left(1-\nu \right)+2\nu {\overline{g}}^{\alpha \beta }{\overline{g}}^{\rho \mu }\right)$

1. The components of the stress-resultant tensor are given by

${T}^{\alpha \beta }=\underset{-h/2}{\overset{h/2}{\int }}\left({\sigma }^{\alpha \beta }-{x}_{3}{\sigma }^{\gamma \alpha }{\kappa }_{\gamma }^{\beta }\right)\left(1+{x}_{3}{\kappa }_{\lambda }^{\lambda }\right)d{x}_{3}$

Substituting the formula for stress components into this expression and integrating through the thickness of the shell gives

${T}^{\alpha \beta }=h{D}^{\alpha \beta \rho \lambda }{\gamma }_{\rho \lambda }+\frac{{h}^{3}}{12}\left({\kappa }_{\mu }^{\mu }{\delta }_{\theta }^{\beta }-{\kappa }_{\theta }^{\beta }\right){D}^{\theta \alpha \rho \lambda }\Delta {\kappa }_{\rho \lambda }-\frac{{h}^{3}}{12}{\kappa }_{\mu }^{\mu }{D}^{\theta \alpha \rho \lambda }{\kappa }_{\theta }^{\beta }{\gamma }_{\rho \lambda }$

Here, terms of order ${h}^{4}$ have been neglected.

1. The components of the internal moment tensor are

${M}^{\alpha \beta }=\underset{-h/2}{\overset{h/2}{\int }}{x}_{3}\left({\sigma }^{\alpha \beta }-{x}_{3}{\sigma }^{\gamma \alpha }{\kappa }_{\gamma }^{\beta }\right)\left(1+{x}_{3}{\kappa }_{\lambda }^{\lambda }\right)d{x}_{3}$

Substituting the formula for stress components into this expression and integrating through the thickness of the shell gives

${M}^{\alpha \beta }=\frac{{h}^{3}}{12}{D}^{\alpha \beta \rho \lambda }\Delta {\kappa }_{\rho \lambda }+\frac{{h}^{3}}{12}\left({\kappa }_{\mu }^{\mu }{\delta }_{\theta }^{\beta }-{\kappa }_{\theta }^{\beta }\right){D}^{\theta \alpha \rho \lambda }{\gamma }_{\rho \lambda }$

where terms of order ${h}^{5}$ and higher have been neglected.

10.5.10 Strain energy and kinetic energy of an elastic shell

It is useful to express the strain energy and kinetic energy of a deformed shell in terms of the motion and deformation of its mid-plane.  To this end:

1. Consider an isotropic, linear elastic shell, with Young’s modulus E,  Poission’s ratio $\nu$, mass density $\rho$ and thickness h.  Denote the contravariant components of the tensor of elastic constants by

${D}^{\alpha \beta \rho \mu }=\frac{E}{2\left(1-{\nu }^{2}\right)}\left(\left({\overline{g}}^{\alpha \rho }{\overline{g}}^{\beta \mu }+{\overline{g}}^{\alpha \mu }{\overline{g}}^{\beta \rho }\right)\left(1-\nu \right)+2\nu {\overline{g}}^{\alpha \beta }{\overline{g}}^{\rho \mu }\right)$

1. Let ${\gamma }_{\alpha \beta }$ and $\Delta {\kappa }_{\lambda \beta }$ denote the covariant components of the mid-plane Lagrange strain tensor, defined in the preceding section,
2. Let $v={v}^{i}{m}_{i}={v}_{i}{m}^{i}$ and $\omega ={\stackrel{˙}{\mu }}_{\alpha }{m}_{3}×{m}^{\alpha }={\stackrel{˙}{\mu }}^{\alpha }{m}_{3}×{m}_{\alpha }$ denote the linear and angular velocity of the mid-plane of the shell.

The total strain energy of the shell can be calculated as

$\Phi =\frac{h}{2}{D}^{\alpha \beta \rho \mu }\underset{A}{\int }\left({\gamma }_{\alpha \beta }{\gamma }_{\rho \mu }+\frac{{h}^{2}}{12}\left(\Delta {\kappa }_{\alpha \beta }\Delta {\kappa }_{\rho \mu }+2{\kappa }_{\lambda }^{\lambda }\Delta {\kappa }_{\alpha \beta }{\gamma }_{\rho \mu }\right)\right)dA$

For all but a very few special shell geometries this result may be approximated by

$\Phi \text{\hspace{0.17em}}\approx \frac{h}{2}{D}^{\alpha \beta \rho \mu }\underset{A}{\int }\left({\gamma }_{\alpha \beta }{\gamma }_{\rho \mu }+\frac{{h}^{2}}{12}\Delta {\kappa }_{\alpha \beta }\Delta {\kappa }_{\rho \mu }\right)dA$

The kinetic energy $K$ can be calculated using the formula

$\text{\hspace{0.17em}}K=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{A}{\int }\left(\frac{h}{2}\rho {v}^{i}{v}_{i}+\frac{{h}^{3}}{24}\rho {\stackrel{˙}{\mu }}^{\alpha }{\stackrel{˙}{\mu }}_{\alpha }\right)dA$

The second term in the expression for the kinetic energy represents the rotational energy. In many practical problems, such as vibration of a shell, the rotational energy can be neglected.

Derivation: The strain energy density in the shell is given by $\Sigma :E/2$, where $\Sigma$ is the material stress tensor and E is the Lagrange strain tensor, defined in the preceding section.  The stress can be expressed in terms of the strain using the constitutive equation, while the strain can be expressed in terms of ${\gamma }_{\alpha \beta }$ and $\Delta {\kappa }_{\lambda \beta }$ using step (3) in Section 10.5.8.   Integrating over the volume of material in the shell, and evaluating the integral through the shell’s thickness explicitly gives the result stated.  The kinetic energy is calculated using the formula for the velocity field in 10.2.8.