This paper discuss the advancement of the PCLD technique used for structural vibration control. In addition to that, there are a lot of sources on the models developed.
[[Viscoelasticity|Viscoelastic materials (VEM)]] dissipate energy under a transient deformation. Used in a form of a layer that is either freely attached (UCLD ie *unconstrained layer damping*) or in a sandwich (CLD/PCLD ie *constrained layer damping/passive constrained layer damping*).
In most of the analyses, extensional/compressional strains of the viscoelastic layer are not taken into account since the damping comes mostly from the shear strain.
The mathematical models are either [[Finite element method|FE]] or analytical.
This paper evaluates the vibrations characteristics of structures with [[Viscoelasticity|viscoelastic materials]]. The equations of motions are derived with the [[Principle of Virtual Displacements|principle of virtual displacement (PVD)]] and solved with the [[Finite element method]].
This paper uses the [[Layerwise Theories|layerwise]] approach to tackle the analysis.
- the solution of nonlinear complex [[Eigenvalues and eigenvectors|eigenvalues]] problems
-> methods have been developed like the [[Modal strain energy technique]], the direct frequency response method, the iterative complex eigensolution and the asymptotic solution method
This paper develops and assess a laminated plate theory x 3D finite element, based on [[Zig-Zag Theories|first order zig zag sublaminate approximations]].
Zig Zag functions are evaluated by enforcing the continuity of the transverse shear stresses at layer interfaces.
=> accounts for discrete layers without increasing the number of degrees of freedom as the number of layers is increased.
5 degrees of freedom per node (8 nodes brick), 3 translation and 2 rotations.
full name : zig-zag in-plane displacement theories
[[Equivalent Single Layer Theories|ESL]] : the laminate is modeled as an equivalent single anisotropic layer
-> most popular : [[Equivalent Single Layer Theories#First-order Shear Deformation Theory (FSDT)|FSDT]] , but does not account for warpage of the cross section.
High-order Shear Deformation theory (HSDT)
[[Equivalent Single Layer Theories#High-order Shear Deformation theory (HSDT)|HSDT]] : it is assumed that the displacements are of higher order polynomial form and are $C^1$ continuous through the thickness. This allows for non-linear variation of displacements, strain and stresses through the thickness.
[[Equivalent Single Layer Theories|ESL]] issue : unable to account for discontinuities in transverse shear strains at interfaces between layers with different stiffness.
[[Layerwise Theories|Layerwise]] : unique displacement field per layer + interlaminar continuity of displacements (and sometimes of transverse stresses).
-> very computationally expensive, since the number of degrees of freedom increase proportionally with the number of layers.
FZZT (First Order Zig-Zag Theory) :
In-plane displacements are assumed to be layerwise linear and continuous through the thickness.
5 degrees of freedom (does not depend on the number of layers) achieved with the transverse shear stress continuity at each interface.
-> very good with symmetrical laminates
HZZT (Higher Order Zig-Zag Theories) :
FZZT + piecewise linear variation of in-plane displacement on a continuous cubic function of the transverse coordinate.
-> better displacement field for unsymmetrical laminates.
\+ homogeneous shear traction boundary conditions at the top and bottom surfaces to keep 5 degrees of freedom.
issue : the transverse deflection degree of freedom $w_0$ is required to be $C^1$ continuous. Therefor 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.
Goal : keep it accurate, $C^0$ continuous and 5 degrees of freedom
This papers explore the history of the development of zig-zag theories, their hypothesis and use-cases. It intends as well to properly address who contributed to what.
[[Zig-Zag Theories]] are theories which describe the piecewise form of transverse stress (Zig-Zag, ZZ) and displacement fields (Interlaminar Continuity, IC).