Dynamic Analysis and Visualization of Laminate Mindlin Plate

Question

I am trying to make the jump from a FEM plot of a static load applied to Mindlin laminate plate to a surface visualization of various Mode Shapes of the Laminate Plate. For that, I understand that the two big steps are:

1. developing a Mass Matrix $[M]$; and

2. getting the eigenvectors and eigenvalues $[V]$ and $[D]$.

While all my textbooks tell me how to do this for a simple Mindlin plate, I don't know how to do it for a laminate Mindlin plate where the very top and bottom layers are of different thickness. All the resources I have found only deal with the most generic of situations. If anyone of you could explain to me what changes (if any) I need to make to the equations, that would be great.

After assembling my mass and stiffness matrices I solve the generalized eigenvalue problem

D, V = eig(stiffness(activeDof, activeDof), M_g(activeDof, activeDof)

But then I do not fully understand what I do with the results. Also, I don't get how to represent the modes in graph form. I know that I'm asking a lot, but if you could at least point me toward a good resource.

Answer 1

Regarding your first question. One option would be to try to use what is commonly done in laminates for Kirchoff-Love plates, I would not know how good is that approximation, though.

Another option, is to consider some effective properties and just replace your constitutive tensor in your equations. For example, for the orthotropic case you would have

$$\begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{12} & C_{22} & C_{23} \\ C_{13} & C_{23} & C_{33} \end{bmatrix} = \frac{1}{1-\nu_{12}\nu_{21}} \begin{bmatrix} E_1 & \nu_{12}E_2 & 0 \\ \nu_{21}E_1 & E_2 & 0 \\ 0 & 0 & 2G_{12}(1-\nu_{12}\nu_{21}) \end{bmatrix} \, ,$$

and in the case of transverse loading in a Kirchoff plate this would lead to

$$D_x \frac{\partial^4 w}{\partial x^4} + 2 D_{xy}\frac{\partial^4 w}{\partial x^2 \partial y^2} + D_y \frac{\partial^4 w}{\partial y^4} = q - \rho h \frac{\partial^2 w}{\partial t^2}$$ where

\begin{align} D_x & = \frac{2h^3 E_1}{3(1 - \nu_{12}\nu_{21})} \\ D_y & =\frac{2h^3 E_2}{3(1 - \nu_{12}\nu_{21})} \\ D_{xy} & = \frac{4h^3 G_{12}}{3} + \frac{2h^3 \nu_{21} E_1}{3(1 - \nu_{12}\nu_{21})} \, , \end{align}

for sure it would be more complex in the case of Mindlin-Reissner plates. You could probably check the references suggested below.

References

1. Leissa, A. W., & Qatu, M. S. (2011). Vibrations of Continuous Systems.McGraw-Hill.

2. Reddy, J. N. (2004). Mechanics of laminated composite plates and shells: theory and analysis. CRC press.