38 lines
2.2 KiB
Markdown
38 lines
2.2 KiB
Markdown
*zig-zag in-plane displacement theories*
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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.
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They are theories which describe the piecewise form of transverse stress (Zig-Zag, ZZ) and displacement fields (Interlaminar Continuity, IC).
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# General Zig-Zag Theories
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([[Secondary articles descriptions#First order Zig-Zag plate Theory|source]])
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[[Layerwise Theories]] with shear continuities between layers to keep the same number of degrees of freedom as the number of layers is increased.
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It captures the "zig-zag" pattern of displacements through the thickness that occurs due to different layer stiffness.
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Improvements of the model consist in keeping it accurate, only needing $C^0$ continuous and 5 degrees of freedom.
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# First Order Zig-Zag Theory (FZZT)
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In-plane displacements are assumed to be layerwise linear and continuous through the thickness. This allows for slope discontinuities at layer interfaces.
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5 degrees of freedom (does not depend on the number of layers) are achieved by using the transverse shear stress continuity at each interface.
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This theory is the most accurate for symmetrical laminates.
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# Higher Order Zig-Zag Theories (HZZT)
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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.
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This allows for better displacement field of unsymmetrical laminates.
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The homogeneous shear traction boundary conditions at the top and bottom surfaces allows us to keep 5 degrees of freedom.
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However the main issue is that the transverse deflection degree of freedom $w_0$ is required to be $C^1$ continuous.
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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.
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# Refined Zig-Zag Theory (RZT) ?
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https://rzt.larc.nasa.gov/ |