next up previous
Next: Elastic Solids Up: Cylindrical Shells Previous: The Membrane Shell

The Naghdi-Cooper System II Formulation

As we mentioned, the membrane model of the cylindrical shell neglects certain important effects, in particular the crucial transverse shear effects. Many so-called higher-order shell theories have appeared in the literature; for a good survey of these theories, we refer to [77,81]. We have decided to focus on the Naghdi-Cooper system II shell model [31,77,128] because it can be simply interpreted as a passive circuit. We note in passing that not all shell theories have this property--Mirsky-Herrmann theory [77], for example, does not, and Naghdi and Cooper's system II, for example, is simplified from their proposed system I which also does not. The problem, more specifically, is that these systems can apparently not be written in the special symmetric hyperbolic form of (3.1), for which the matrices $ {\bf A}_{1}$ and $ {\bf A}_{2}$ will be independent of $ x$ and $ \theta$. This property is essential here in that such dependence considerably complicates the inter-loop coupling in an MDKC (notice that the port-resistances of the Jaumann two-ports which realize this coupling been constant for every system we have looked at so far).

Another reason for choosing this particular shell model is that it can be simply written as Mindlin's plate system (in coordinates $ x$ and $ \theta$ instead of $ x$ and $ y$) coupled with the membrane shell. We can write this system in the form of (3.1), where the dependent variable $ {\bf w}$ is defined by

$\displaystyle {\bf w} = [v_{z}, q_{x}, q_{\theta}, \omega_{x}, \omega_{\theta}, m_{x}, m_{y}, m_{xy}, v_{x}, v_{\theta}, n_{x}, n_{\theta},n_{x\theta}]^{T}$ (5.53)

Here, the first 8 variables are precisely those that appear in the Mindlin theory (see §5.4), but in cylindrical coordinates; we have changed subscripts $ y$ to $ \theta$, and written the $ v_{z}$ instead of $ v$ for the radial transverse velocity. The other five variables appear in subsystem (5.47) of the membrane shell theory (see §5.5.1). The system matrices are
$\displaystyle {\bf P} = {\bf P}_{NC} = \begin{bmatrix}
{\bf P}_{M}&\cdot\\
\cdot&{\bf P}_{R}^{-}\\
{\bf A}_{1}$ $\displaystyle =$ $\displaystyle {\bf A}_{NC1} = \begin{bmatrix}
{\bf A}_{M1}&\cdot\\
\cdot&{\bf ...{bmatrix}
{\bf A}_{M2}&\cdot\\
\cdot&{\bf A}_{M2}^{-}\\
$\displaystyle {\bf B}$ $\displaystyle =$ $\displaystyle {\bf B}_{NC} = \begin{bmatrix}
{\bf B}_{M}&{\bf B}_{NC\times}\\
-{\bf B}_{NC\times}^{T}&\cdot\\

where $ {\bf P}_{M}$, $ {\bf A}_{M1}$, $ {\bf A}_{M1}^{-}$, $ {\bf A}_{M2}$, $ {\bf A}_{M2}^{-}$ and $ {\bf B}_{M}$ are as defined in §5.4, and $ {\bf P}_{R}^{-}$ appears in §5.5.1. The coupling matrix is

$\displaystyle {\bf B}_{NC\times} = \begin{bmatrix}0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0...
...0&0&0&0&0&0&0\\ \frac{1}{a}&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ \end{bmatrix}^{T}$    

Note that this coupling disappears in the limit as the shell radius $ a$ becomes large (effectively leaving us with with the Mindlin system). $ {\bf B}_{NC}$ is indeed anti-symmetric, so we are guaranteed a lossless MDKC network; in fact, this network can be directly constructed from the two networks shown in Figures 5.16 and 5.23, by attaching terminals $ A$ and $ A'$ in the former to $ B$ and $ B'$ in the latter. The scaling parameters $ r_{1}$, $ r_{2}$ and $ r_{3}$ can be chosen optimally according to (5.37), (5.40) and (5.41), and $ r_{4}$ and $ r_{5}$ can be set as in (5.48) and (5.49), giving a bound for passivity on the combined network,

$\displaystyle v_{0}\geq v_{NC}\triangleq\max(v_{M},v_{R})$    

where $ v_{M}$ is the bounding space step/time step ratio for the Mindlin network, from (5.43) (in cylindrical coordinates), and $ v_{R}$ is the same quantity for the membrane shell system, from (5.50).

Figure 5.24: Computational grid for the DWN for Naghdi and Cooper's system II. Grids (a) and (b) correspond to a DWN for a Mindlin-type subsystem, and are coupled to a membrane shell-type DWN operating on grid (c). Grid functions (capitalized versions of the dependent variables (5.51)) are indicated next to the grid points at which they are calculated. Grey/white coloring of grid points indicates calculation at parallel/series junctions at alternating time steps.
\end{picture} \end{center}\par\vspace{0.4in}

Because this system can be constructed entirely by connecting subnetworks that we have already examined in detail, it seems unnecessary to show the discrete MDWD network or the alternate MDKC and its discrete form suitable for DWN implementation. The MDWD network will be exactly the combination of the ``Mindlin'' system, shown at bottom in Figure 5.16, and the MDWD network corresponding to the MDKC for the membrane shell, shown in Figure 5.23; recall that the MDKC for the membrane shell system (with a free port with terminals $ B$ and $ B'$) is identical in form to that of the uncoupled five-variable Mindlin subsystem, and thus its MDWD counterpart will be of the same form as well.

For the transformed network to be used to generate a DWN, a few comments are in order. For the Mindlin system, we first applied network theoretic rules in order to arrive at a modified form, shown at top in Figure 5.17. In this case, the transverse velocity $ v$ (renamed $ v_{z}$ in this section) and the bending moments $ m_{x}$, $ m_{y}$ and $ m_{xy}$ (renamed $ m_{x}$, $ m_{\theta}$ and $ m_{x\theta}$) have been interpreted as voltages instead of currents. This transformed network can then be connected (via terminals $ A$ and $ A'$ in Figure 5.17) to a transformed form of the membrane shell system, shown in Figure 5.23; for the shell subsystem, $ n_{x}$, $ n_{\theta}$ and $ n_{x\theta}$ will be treated as voltages, and $ v_{x}$ and $ v_{\theta}$ as currents.

After connecting these transformed subnetworks, and applying the usual alternative discretization rules, we end up with a DWN that will operate on an interleaved grid as shown in Figure 5.24. Grids (a) and (b) are precisely the Mindlin grid shown in Figure 5.18, and are coupled instantaneously to a third grid (c), which adds the effect of curvature to the system.

next up previous
Next: Elastic Solids Up: Cylindrical Shells Previous: The Membrane Shell
Stefan Bilbao 2002-01-22