Micro/macro analysis of a compositeCasa Day 22/11/06

Aloha Airlines Flight 243

Motivation

Honolulu International Airport, Hawaii

At 24000 feet… loud clap

No visual exterior inspection (nor required)

Flight attendant swept out of the cabin

1 dead, 65 injured

$5 million damage

Peel Test

45º

Force Displacement (constant speed)

Nickel layer

Substrate

Peel-Test Support

Composites

Matrix

Inclusions

Elasticity problem for a composite material

Matrix

InclusionsMatrix Inclusions

Material Interface

Modelling

Matrix

InclusionsMatrix Inclusions

Material Interface

Homogenisation

Homogenisation

fu Α fu Α

0

uu

Homogenisation

Homogenised equations

Boundary Conditions

Macro results

Micro/Macro

Domain decomposition

Dirichlet – Neumann scheme

Finite element approximation

Finite element approximation

Schur complement system

• symmetric

• positive definite

• ill conditioned

Algorithm

Dirichlet problem

Neumann Problem

Relaxation step

Algorithm

is a preconditioner for

Theorem:

This process converges if

where is the maximum eigenvalue of .

Heterogeneous domain decomposition

Heterogeneous domain decomposition

Interface conditions

Overview

homoge-

nisation

D-N

Interface conditions?

Example

Reference solution

Macro domain

Meso domain

Example

Macro domain

Meso domain

Absolute error for

the horizontal displacementu1-uref 0.6012

u2-uref 0.6701

u3-uref 0.1392

u4-uref 0.0305

u5-uref 0.0176

Future Work

- Interface conditions discussion

- Crack propagation