Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007 Cellular Materials: Structure and Properties M. Emília Rosa ICEMS-Instituto de Ciência e Engenharia de Materiais e Superfícies Departamento de Engenharia de Materiais Instituto Superior Técnico, Universidade Técnica de Lisboa Av. Rovisco Pais, 1049-001 Lisboa, Portugal
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Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Cellular Materials: Structure and Properties
M. Emília Rosa
ICEMS-Instituto de Ciência e Engenharia de Materiais e SuperfíciesDepartamento de Engenharia de Materiais
Instituto Superior Técnico, Universidade Técnica de LisboaAv. Rovisco Pais, 1049-001 Lisboa, Portugal
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
DEFINITIONS
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
CELLULAR MATERIAL: assembly of cells with solid edges and faces,packed together to fill space.
CELL (Robert Hooke, 1660) derives from Latin cella.
CELLA: small compartment; an enclosed space.
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Robert Hooke, 1660: CORK was one of the first materials he examined at his microscope.
Cork cellular material
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
TYPES of cellular materials
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Cellular materials are very common in Nature
Cork and Balsa
Sponge and Cancellous Bone
Coral and Cuttlefish Bone
Iris Leaf and Stalk of a Plant
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Man makes 3 dimensional (3D) cellular materials FOAMS
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
2D
(each edge is shared between 2 vertices)
From Euler’s law
If F is large
For any
3=eZ 23=VE
sides withfaces of number nFn = ∑ = EFn n
2
FFFn n 66 =
− ∑
6==∑ nF
Fn n
eZ2
2−
=e
e
ZZ
n
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
3D
An isolated cell (C=1)
For any
3=eZ
−=
fn 216
14.514512
====
nfnf
−
−=
fZZZ
nZe
fee
212
Zand f
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Dispersion of cell size
Competitive growth
Voronoi honeycomb (2D) or foam (3D)Bubbles nucleate randomly in space at the same time and all grow with thesame linear growth rate characteristic cell centred on the point of nucleation and contains all points which are closer to this nucleation pointthan to any other.
Random Voronoi honeycomb and Voronoi honeycomb for a set of points initially random, from which all points closer than a critical spacing were removed
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Surface tension (T)
Minimizes surface area at constant cell volume; cell edges in a honeycomb and cell faces in a foam meet at 120º; faces have a curvature which isrelated to the pressure difference Δp between the pair of cells which meetat the faces.
Coarsening of cells
Important in soap foams and during foaming of polymers.Gas or fluid in one cell diffuses through its walls into the surrounding cells.
D)3(11D)2(21
+==
rrTp
rTp ∆∆
( ) ( )( )( )
( )( ) D)3(D)2(
D)3(ddD)2(6
dd
0
0
0
0
21
ffff
fVfV
nnnn
nAnA
ffCtVnC
tA
−−
=−−
=
−=−=
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
2D
A cell with more sides than average has neighbours which, taken together,have less than the average number.
3D
5763 =⇒==⇒= nnnZe
nm 65 +=
fg 1413 +=
121614 =⇒== fff
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Anisotropy
Three orthogonal sections of an anisotropic polyurethane foam
Sρρ
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Relative density ( )
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Characterization chart for honeycombs
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Characterization chart for foams
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
MECHANICS of honeycombs
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Compression and tension stress-strain curves
Elastomeric Honeycomb
Elastic-plastic Honeycomb
Elastic-brittle Honeycomb
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Schematic stress-strain curves of compact materials
Metals are elastic-plastic PolymersCeramics are elastic-brittle
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
Model of a honeycomb with hexagonal cells
In-plane properties are those relating to loads applied in the X1-X2 plane; responses to loadsapplied in the X3 direction are referred to as the out-of-plane properties
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007
In-plane deformation
Elastic Bending
( )
12221121
2
1
212
3
3
S
2
2
3
S
1
21
3
S
3
S
3
21
1
sinsin
cos
cos
sin
sinsinhcos
sinhcos
cossin
1212cos
12sin
2cos
2sin
cossin
nnnqq
qeen
q
q
qq
q
qqde
qqde
qdqd
qq
qsqs
EEh
ht
EEt
EE
tbIIE
WIE
P
WMPM
bWbhP
==÷øöç
èæ +
=-=
÷øöç
èæ +
÷øöç
èæ=
÷øöç
èæ +
÷øöç
èæ=
+==
===
==
=+=
l
l
l
l
l
ll
ll
ll
ll
Marie Curie School on Knowledge Based Materials – Estremoz, Portugal – 21.August.2007