*zig-zag in-plane displacement theories*

To [[Composite laminate models|model a composite laminate]] there are a few options, the most important are the [[Equivalent Single Layer Theories|equivalent single layer]] and the [[Layerwise Theories|layerwise]]. However there was a need to get theories that were less computationally expensive than the layerwise, and more accurate than the equivalent single layer.

They are theories which describe the piecewise form of transverse stress (Zig-Zag, ZZ) and displacement fields (Interlaminar Continuity, IC).

# General Zig-Zag Theories
([[Secondary articles descriptions#First order Zig-Zag plate Theory|source]])

[[Layerwise Theories]] with shear continuities between layers to keep the same number of degrees of freedom as the number of layers is increased.

It captures the "zig-zag" pattern of displacements through the thickness that occurs due to different layer stiffness.

Improvements of the model consist in keeping it accurate, only needing $C^0$ continuous and 5 degrees of freedom.

# First Order Zig-Zag Theory (FZZT)

In-plane displacements are assumed to be layerwise linear and continuous through the thickness. This allows for slope discontinuities at layer interfaces.

5 degrees of freedom (does not depend on the number of layers) are achieved by using the transverse shear stress continuity at each interface.

This theory is the most accurate for symmetrical laminates.

# Higher Order Zig-Zag Theories (HZZT)

To improve the [[#First Order Zig-Zag Theory (FZZT)|FZZT]], a piecewise linear variation of in-plane displacement is superimposed on a continuous cubic function of the transverse coordinate.
This allows for better displacement field of unsymmetrical laminates. 

The homogeneous shear traction boundary conditions at the top and bottom surfaces allows us to keep 5 degrees of freedom.

However the main issue is that the transverse deflection degree of freedom $w_0$ is required to be $C^1$ continuous. 

Therefore additional rotational degrees of freedom (gradients of $w_0$) are present -> more than 6 degrees of freedom -> tough to implement in commercial finite element software.


# Refined Zig-Zag Theory (RZT) ?

https://rzt.larc.nasa.gov/