materiais compósitos propriedades mecânicas dos materiais
TRANSCRIPT
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M
Materiais Compósitos
Propriedades mecânicas dos materiais
Micro – mecânica
Micromechanics
Departamento de Engenharia Mecânica
Instituto Superior Técnico
Materiais Compósitos
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MMicro / macromecânica
The term “micromechanics” does not refer to
mechanical behavior at the molecular level.
Looks at components of a composite, the matrix
and the fiber, and tries to predict the behavior of
the assumed homogeneous composite material.
The behavior of the lamina is called
“macromechanics”.
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MMicro / macromecânica
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MPropriedades mecânicas das fibras (1)
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MFibras em lâminas
• Alguns arranjos típicos de fibras em cada camada de compósito
a - Fibras unidireccionais contínuas
b - Fibras descontínuas orientadas de modo aleatório
c - Fibras unidirecionais tecidas ortogonalmente
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MTipos de tecidos
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MMatrix Properties
• 1. Good mechanical
properties
• 2. Good adhesive
properties
• 3. Good toughness
properties
• 4. Good resistance
to environmental
degradation
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MMechanical properties
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MTipos de resina (2)
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MFiber properties
1. High resistance
2. Linear elastic behaviour
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MMicro-mecânica - FracçãoVolumica
composite of volume total
component i theof volume
component i theoffraction volume
1
th
th
i
i
1
i
V
V
v
V
Vv
v
i
i
n
i
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MMicro-mecânica - FracçãoVolumica
f m v
volume fraction of the fiber
volume fraction of the matrix
volume fraction of the voids
f
f
m
m
v
v
v v v 1
Vv
V
Vv
V
Vv
V
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MMicro-mecânica – Fracção mássica
th
th
weight fraction of the i component
weight of the i component
total weight of composite
n
i
i 1
i
i
i
i
w 1
Ww
W
w
W
W
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MMicro-mecânica – Fracção mássica
weight fraction of the fiber
weight fraction of the matrix
weight fraction of the voids
f m
f
m
v
w w 1
w
w
w 0
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MMicro-mecânica – densidade
composite ofweight total
component i the of volume
component i the ofweight
component i the of densityweight
i
th
th
thi
ii
i
i
i
i
VWW
V
W
V
W
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MMicro-mecânica – densidade
th
i
weight density of the composite
weight of the i component
volume of the composite
weight density of the composite
c
c
c
n
c i
i 1
W
V
W W
V
v
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MMicro-mecânica – densidade
• Autoclave cured: P = 0.1 – 1%
• Only Vacuum bagging: P = 5% (approx.)
– Regra das Misturas (Rule of Mixtures)
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MPropriedades da camada ortotrópica
Area=1 m2
if
of
i
fof
fh
m
hm
m
volumeComposite
volumeFiberV
..1
/2
ih
ff
iV
gramagemh
mof - Gramagem - Fiber
weight / square meter
(grams/m2)
hi – espessura da camada /
ply thickness (mm)
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MPropriedades da camada ortotrópica
n
hh ti
o90
o0
o90
o30
o45
o45
th
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MEspessura das lâminas / Ply and laminate thichness
n
hh
V
mh
t
i
ff
of
i
Plano de Simetria
- 45°
90°
+ 45°
0°
mof - Gramagem - Fiber
weight / square meter
(grams/m2)
hi – espessura da
camada / ply thickness
(mm)
ht – espessura total /
laminate thickness
(mm)
n – camadas / nº of
layers
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MLaminate Orientation Code
• Ply angles given in degrees - 45 or -45
• Separated by slashes - 0/90
• Listed top to bottom layer
• Enclosed in square brackets - [0/90/0]
• Subscripts used– s – symmetric laminate top half given
– (0)n – repeat layer n times
• Center layer uses overbar in odd number layer symmetric laminate
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M
s45/90/0
o0
o0
o90
o90
o45
o45
o0
o0
o30
o30
o30
o30
Laminate Stacking Sequence Examples
q 30 o
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M
o45
o90
o45
o0
o0
o90
o0
o0
o45
o45
s2 45/90/0
Laminate Stacking Sequence Examples
s2 90/0/45
o90
o0
o0
o90
o0
o0
o90
o90
o45
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M
45/90/0/30/45/90
Laminate Stacking Sequence Examples
o45
o45
o0
o0
o45
o45
o0
o0
o90
o90
o0
o90
o30
o45
o45
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MComportamento da Lâmina
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MComportamento da Lâmina
LT
LTLT
L
TLT
L
L
LT
T
TT
T
T
TL
L
LL
G
EE
EE
)1(2
EG
G
EE
E
ν
E
xy
xy
x
y
x
y
y
yx
x
Isotropic MaterialOrthotropic Material
Material ortotrópico: Material com
3 planos de simetria mutuamente
perpendiculares.
Orthotropic material : Material with
three mutually perpendicular
planes of symmetry .
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MComportamento da Lâmina
Isotropic Material versus orthotropic Material
Orthotropic material : Material with
three mutually perpendicular
planes of symmetry . Usually,
e1ǂe2ǂe3
Material ortotrópico: Material com
3 planos de simetria mutuamente
perpendiculares.
Usualmente, e1ǂe2ǂe3
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MEquações constitutivas – materiais ortotrópicos
=
2
23
3
32
1
13
3
31
1
12
2
21
12
13
23
33
22
11
12
13
23
32
23
1
13
3
32
21
12
3
31
2
21
1
12
13
23
33
22
11
;;
com
100000
01
0000
001
000
0001
0001
0001
EEEEEE
G
G
G
EEE
EEE
EEE
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MEquações constitutivas – materiais ortotrópicos
=
Eq. 4.7
3
3113
ES
111
1
ES
12
13
23
33
22
11
12
13
23
32
23
1
13
3
32
21
12
3
31
2
21
1
12
13
23
33
22
11
100000
01
0000
001
000
0001
0001
0001
G
G
G
EEE
EEE
EEE
2
2323
ES
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En
ge
nh
ari
a d
e M
ate
ria
isComportamento da Lâmina
Lets consider one lamina in 1-2 (x-y) plane, (Z=3)
as shown in Figure 3.1. Then:
Based on this, the following terms become:
And:
1
12
2
21
EE
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MDefinitions
Transverse
means
perpendicular to
the fibers or in the
T (2) - direction
Longitudinal
means in the fiber
direction or in the
L (1) - direction shear
stress transverse
stress allongitudin
)(
)(
LT
T
L
strainshear
strain transverse
strain allongitudin
LT
T
L
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MPropriedades da camada ortotrópica
n
hh
V
gramagemh
ti
ffi
LT
T
L
LT
TL
LT
T
TL
L
LT
T
L
G
EE
EE
100
01
01
T
TL
L
LT
EE
mof - Gramagem - Fiber
weight / square meter
(grams/m2)
hi – espessura da camada /
ply thickness (mm)
ht – espessura total /
laminate thickness (mm)
n – camadas / nº of layers
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M
LTLTLT
T
LLT
TTT
LLL
G
E
E
Assume Linear Behavior
121212
2
112
222
111
G
E
E
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M
matrix
matrix
fiber L
T
areamatrix
areafiber
area total
m
f
A
A
AAf
Representative Volume Element (RVE)
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M
1c1c
Micro-mecânica (1)
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MMicro-mecânica (1)
mLfLL
ffmmL VEVEE
mmLffLLL AAAF
A
AE
A
AEE
A
A
A
Am
mLmL
f
fLfLLL
m
mL
f
fLL
mf AAA
f – fibra ; m – matriz
L – direcção das fibras
T – direcção transversal às fibras
Força na direcção das fibras - Isostrain
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MMicro-mecânica (1)
Lfmfm hhh
fh
2mh
2mh
L
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MMicro-mecânica (1)
Para o mesmo tipo de carregamento e os mesmos materiais
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MMicro-mecânica (1)
L
fmfm hhh
fh
2mh
2mh
L
Am.hm = Matrix Volume
Ac.hc = Composite Volume
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MMicro-mecânica (1)
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MMicro-mecânica (2)
• Força na direcção
perpendicular às
fibras - Isostress
mTfTT
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M
2c
2LfL
2Lm
2Lm
Micro-mecânica (2)
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MMicro-mecânica-Isostress
2c
2LfL
2Lm
2Lm
mTfTT
As we see earlier
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MMicro-mecânica-Isostress
2c
2LfL
2Lm
2Lm
Lm.Am = Matrix Volume
Lc.Ac = Composite Volume
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MMicro-mecânica-Isostress
2c
2LfL
2Lm
2Lm
So
m
m
f
f
T
m
m
mT
f
f
fT
T
T
mmTffTT
c
c
E
V
E
V
EV
EV
EE
VVl
l
1
mTfTT
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MMicro-mecânica (3)
• Exemplo da variação do
modulo de elasticidade de
um compósito (Ec) de fibra
de vidro e resina poliester
em função da % vol. de
fibra (Vf) e dos respectivos
módulos Ef (fibra) e Em
(matriz)
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MMicro-mecânica (3)
• Example of variation of the
modulus of elasticity of a
fiberglass and polyester
resin composite (Ec) as a
function of volume % (Vf)
and their modules Ef (fiber)
and Em (matrix).
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M
Micro-mecânica - Shear modulus
2m
f
2m
mfc
c
cc
f
ff
m
mm
G
G
G
2/~ Ltg
2L
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MMicro-mecânica - Shear modulus
Shear Modulus of fiber in 12-plane
Shear Modulus of matrix
f m
12 f 12 m
f 12
m
v v1
G G G
G
G
mfc
c
cc
f
f
f
m
mm
mfc
G
G
G
AAA
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MMicromecânica - final
mmffLT VV Coef. de Poisson
m
m
f
f
T E
V
E
V
E
1Módulo de elasticidade
transversal
m
m
f
f
LT G
V
G
V
G
1Módulo de rigidez ao corte
ffmmL VEVEE Módulo de elasticidade
longitudinal
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MMicromecânica - final
mmffLT VV Poisson Coefficient
m
m
f
f
T E
V
E
V
E
1Transversal Young Modulus
m
m
f
f
LT G
V
G
V
G
1Shear modulus
ffmmL VEVEE Longitudinal Young Modulus
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MMicromecânica - final
• Good values of EL and LT
– These properties appear to be independent of
fiber packing geometry.
– Rule of Mixtures works Well for Fiber Dominated
Properties (EL and LT)
• Less acceptable values for ET and GLT
– These properties appear to be highly dependent
on fiber packing geometry
– Better Models Needed for Matrix Dominated
Properties (E2 and G12)
– Improved models are available
Fibers Properties
53Composite Materials
Short fibers versus Continuous Fibers
Ec = Ef Vf + Em (1-Vf )Continuous Fibers
Short fibers Empirical approach (h1 )
Ec = h1Ef Vf + Em (1-Vf )
Fibers Properties
54Composite Materials
Short fibers versus Continuous
Fibers
Short fibers Empirical approach (h1 ) Ec = h1Ef Vf + Em (1-Vf )
For l ≈10 mm h1 = 0,99
Ec ≈ Ef Vf + Em (1-Vf ) Equation for Long Fibers
Fibers Properties
55Composite Materials
Short fibers versus Continuous Fibers
New Empirical approach (ho )
Ec =ho h1Ef Vf + Em (1-Vf )
When there is a distribution of fiber orientation the reinforcing efficiency
of the fibers reduced.
where ho is an orientation efficiency factor.
For unidirectional laminae ho = 1 when tested to the fibers,
ho = 3/8 for in-plane random fiber distributions and
ho = 1/5 for three-dimensional random distributions
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MComportamento da Lâmina
Let`s come back to the Isotropic material
versus Orthotropic material behavior
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MComportamento da Lâmina
Isotropic material versus Orthotropic material
Orthotropic material : Material with
three mutually perpendicular
planes of symmetry . Usually,
e1ǂe2ǂe3
Material ortotrópico: Material com
3 planos de simetria mutuamente
perpendiculares.
Usualmente, e1ǂe2ǂe3
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MComportamento da Lâmina
LT
LTLT
L
TLT
L
L
LT
T
TT
T
T
TL
L
LL
G
EE
EE
)1(2
EG
G
EE
E
ν
E
xy
xy
x
y
x
y
y
yx
x
Isotropic material Orthotropic material
Material ortotrópico: Material com 3
planos de simetria mutuamente
perpendiculares.
Orthotropic material : Material with
three mutually perpendicular planes
of symmetry . Usually, e1ǂe2ǂe3
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MPropriedades da camada ortotrópica
LT
T
L
LT
TL
LT
T
TL
L
LT
T
L
G
EE
EE
100
01
01
T
TL
L
LT
EE
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M
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M
ORIENTAÇÃO DAS
FIBRAS
A resistência será
máxima quando as
fibras estiverem
orientadas com o
esforço (sendo mínima
na direcção
perpendicular)
Constantes elásticas segundo qualquer direcção - exemplo
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MPropriedades da camada ortotrópica
(L) 1
y
x
(T) 2
q
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MPropriedades da camada ortotrópica
1
y2
qx
q
q sindA2
q cosdA1
dAx
dAxy
q sin12dA
q cos12dA
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MEquilibrium
0cossin
cossincossin
0cossin2
sincos
22
12
21
12
2
2
2
1
qqqq
dA
dAdAdAF
dA
dAdAdAF
xyy
xx
q
q sindA2
q cosdA1
dAx
dAxy
q sin12dA
q cos12dA
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MStress Transformation
Similar derivation for y
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MTransformation in Matrix Form
12
2
1
22
22
22
sincossincossincos
sincos2cossin
sincos2sincos
qqqqqq
qqqq
qqqq
xy
y
x
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MCondensed Matrix Form
12
2
11
T
xy
y
x
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MTransformation Matrix: [T]
xy
y
x
xy
y
x
TT
12
2
1
12
2
11
or
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MMatrices
22
22
22
22
22
22
1
2
2
2
2
sccscs
cscs
cssc
T
sccscs
cscs
cssc
T
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MStrain
xy
y
x
T
12
2
1
12
2
1
1
T
xy
y
x
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MMatrices
22
22
22
22 sccscs
cscs
cssc
T
22
22
22
1
22 sccscs
cscs
cssc
T
Attention, matrices T are different from the matrices T
Ma
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Co
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ós
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DE
M
xy
y
x
xy
y
x
T
Q
T
12
2
1
12
2
1
12
2
1
12
2
1
1
LT
T
L
LT
TL
LT
T
TL
L
LT
T
L
G
EE
EE
100
01
01
General Stress-Strain Behavior
12
2
1
1
12
2
1
Q
Ma
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ais
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ós
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am
ina
do
s
DE
MGeneral Stress-Strain Behavior
xy
y
x
xy
y
x
TQT
1
Ma
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ais
Co
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ós
ito
sL
am
ina
do
s
DE
M
TQTQ1
xy
y
x
xy
y
x
Q
__
Ma
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ais
Co
mp
ós
ito
sL
am
ina
do
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MStress-Strain Behavior – outro formulário
xy
y
x
xy
y
x
QQQ
QQQ
QQQ
662616
262212
161211
Ma
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ós
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do
s
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MStress-Strain Behavior
TQTQ111
xy
y
x
xy
y
x
T
Q
T
12
2
1
12
2
1
1
12
2
1
12
2
1
1
TQTQ1
xy
y
x
xy
y
x
t
TQT
11
L (1), T (2)
xy
y
x
xy
y
x
t
Q
1__
Ma
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ais
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ito
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ina
do
s
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M
• Fim/End