#description

# Collisions
## Hard and Soft Collisions
*webpage*
[ref](https://www.compadre.org/Physlets/mechanics/illustration8_3.cfm)

A soft [[Impact-Shock-Collision|collision]] is an inelastic collision, which means that kinetic energy 
is not conserved (because internal friction). However, the momentum is still
conserved.

A perfectly inelastic collision occurs when the two bodies stays together, 
and the energy is lost by bonding the two bodies.

## Inelastic impacts for composite materials
*1990*
[online ref](https://doi.org/10.1016/0263-8223(90)90025-A)
[[Inelastic_impacts_for_composite_materials.pdf|local ref]]

Use the [[Finite element method|finite element model]] to get the dynamic response.

The structure is considered elastic but the loading is considered
inelastic.

The isoparametric linear shell element is modified
to take into account the shear deformation and rotatory inertia.

[[Inelastic Collisions]] : masses added together and
momentum conserved, used for slamming or other wave-like loading

This model is reasonable when the impactor is relatively soft and the mass 
of impactor is larger than the mass of the node being impacted.

## FEM of low-velocity impact on composite materials
*2011*
[online ref](https://doi.org/10.1016/j.compstruct.2010.10.003)
[[FEM_of_low-velocity_impact_on_composite_materials.pdf|local ref]]

use of ABAQUS

examination of the validity of different models

propose a  benchmark method in low-velocity [[Impact Models|impact modeling]] of composite structures

---
# Background

## Machine Vibration
*2021 (info from the metadata of the PDF)* 
[online ref](https://www.machinedyn.com/docs/articles/Real_Physics.pdf)
[[Real_Physics_of_Machine_Vibration.pdf|local ref]]

Very practical paper, an introduction to vibrations in engineering. It has useful definitions and explanations of the terms.
Critique of Newtonian physics ($F=ma$) and Hooke's law ($F=kx$).
Because it assumes constant masse and stiffness.

In a dynamic world, $F = mr\omega^2$. 

To preserve the linearity of Newton's $2^{nd}$ law a dynamic mass is defined : 
$m(\omega)$.
The reciprocal of dynamic mass is accelerance, and is also a function of frequency : Accelerance $= \frac{1}{m(\omega)} = \frac{a(\omega)}{F(\omega)}$

This paper advocate for less design and more tests because the theory is too far from the real world.

**Symmetry is bad practice because it support resonant modes**

Force is a wave that travels at the speed of sound.

(see [[Accelerance]])

---

# VEM

## Passive Constrained Layer Damping, SotA
*2019*
[online ref](https://iopscience.iop.org/article/10.1088/1757-899X/653/1/012036)
[[PCLD_SotA.pdf|local ref]]

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.

(see [[PCLD]])

## Layerwise Analyses VEM
*2016*
[online ref](https://doi.org/10.1115/1.4034023)
[[Layerwise_Analysis_VEM.pdf|local ref]]

This paper evaluates the vibrations characteristics of structures with [[Viscoelasticity|viscoelastic materials]]. The equations of motions are derived with the principle of virtual displacement (PVD) and solved with the [[Finite element method]].
This paper uses the layerwise approach to tackle the analysis.

This paper focus its study on beams.

Layerwise approach : Lagrange-like polynomial expansions have been adopted to develop the kinematic assumptions (?)

Issues of [[Viscoelasticity|viscoelastic]] layers dynamic study :
- the modeling of material properties
	-> tests to characterize the material
- the solution of nonlinear complex eigenvalue 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
- the kinematic modeling of the structure
	-> main topic of the paper
	-> damping through maximizing shear => need accurate stress distribution

This paper wish to provide an alternative to the 3D modeling, preserving the numerical efficiency of 1D theories.

## Analysis of the compressible neo-Hookean model
*2023*
[online ref](https://link.springer.com/article/10.1007/s11012-022-01633-2)
[[Analysis_of_the_compressible_neo-Hookean_model.pdf|local ref]]

Analysis of the model implemented in the commercial [[Finite element method|finite element]] software ABAQUS, ANSYS and COMSOL.

Its physical limitations are explored, to underline the model's advantages and limitations.

**To further read, but not necessary at first glance for my study**

## Optimization method of composite laminates with a viscoelastic layer
*2009* #perforated_vel #unfinished_read
[[Optimization_of_composite_laminate_with_viscoelastic_layer.pdf|local ref]]

This paper develops a method to transform a multi objective (damping and stiffness)
into a single one, to facilitate optimization.

The [[Viscoelasticity|viscoelastic]] layer is **perforated**, and the sandwich is [[Co-curing|co-cured]].

Co-curing means the viscoelastic material within the composite laminate undergo the temperature and pressure cycle needed to cure the composite material.


---

# Composite laminates

## First order Zig-Zag plate Theory
*2000*
[online ref](https://doi.org/10.1016/S0263-8223(99)00063-X)
[[First_order_Zig-Zag_plate_Theory.pdf|local ref]]

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

(see [[Zig-Zag Theories]])

## Historical review of Zig-Zag theories
*2003*
[online ref](https://asmedigitalcollection.asme.org/appliedmechanicsreviews/article-abstract/56/3/287/446373/Historical-review-of-Zig-Zag-theories-for)
[[Historical_review_of_Zig-Zag_Theories.pdf|local ref]]

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).

This papers explain thoroughly the different theories developed and how they function (maybe a bit too much for what I need).