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COMPOSITE MATERIALS. APPLICATION
introduction
Mechanics of Composite Materials
T h l
Technology of Polymers and Composites & Engineering Mechanics
fP l
dC
i &E i
i M h i
Dmytro Vasiukov
Dmytro
y Vasiukov
dmytro.vasiukov@mines‐‐douai.fr
dmytro.vasiukov@mines
2
COMPOSITE MATERIALS
COMPOSITE MATERIALS
Some videos:
A380 production
BMW i3 production
BMW i3 crash test
http://leehamnews.com/category/american‐airlines
•
3
•
http://www.boeing.com
4
•
http://www.compositesworld.com/
BMW i3 car body
COMPOSITE MATERIALS. APPLICATION
COMPOSITE MATERIALS
introduction
Aerospace
CFRP
introduction
Composite material= combination of two or more materials with significantly different
behavior on a macroscopic scale i e a material with two or more distinct phases namely
behavior on a macroscopic scale, i.e. a material with two or more distinct phases, namely
the reinforcements and matrices Not like an alloy, like e.g. Ti6AlV4, or Al 7050!
Wing Skin, Front Fuselage, Control Surface Fin & Rudder, Access Doors, Under Carriage Doors, Engine Cowlings,
M i Torsion
Main
T i Box,
B
F l Tanks,
Fuel
T k Rotor
R t Blades,
Bl d Fuselage
F l
St t
Structures
and
d Floor
Fl
B d off Helicopters,
Boards
H li t
A t
Antenna
Dishes, Solar Booms and Solar Arrays, etc.
BFRP
Horizontal and Vertical Tail, Stiffening Spars, Ribs and Longerons, etc.
KFRP
Nose Cones, Wing Root, Fairings, Cockpit and Fuselage of Helicopters, Motor Casings, Pressure Bottles, Propellant
Reinforcements can be: particulates (e.g. SiC), whiskers or fibers, embedded in continuous
phases („matrices“), here: restriction to composites with long fibers („advanced“
composites), often abbreviated as FRP (fiber reinforced plastics)
Fiber, monofilament: Primary reinforcement, short (= staple) or long (= continuous), fiber
„size“ Ø =7 ...150 μ
i “ Ø 7 150
Tanks, Other Pressurised Systems, etc.
GFRP
Structural
GFRP
Floor Boards, Interior Decorative Panels, Partitions, Cabin Baggage Racks and Several Similar Applications.
Folded Plates of Various Forms, Both Synclastic and Anticlastic Shells, Skeletal Structures, Walls and Panels,
Matrices: polymers, metals or ceramics; we shall consider polymeric matrices
Doors, Windows, Ladders, Staircases, Chemical and Water Tanks, Cooling Towers, Bridge Decks, Antenna Dishes,
etc.
etc
In general, the reinforcements are much stronger and stiffer than the matrix and matrices
are more ductile than fibers
Marine and Mechanical
GFRP
Ship and Boat Hulls, Masts, Automobile Bodies, Frames and Bumpers, Bodies of Railway Bogeys, Drive shafts,
g Rods,, Suspension
p
Systems,
y
, Instrument Panels.
Connecting
Sports
GFRP/CFRP
Lamina ply laminae: unidirectional (UD) layer is basic element thicknesses (in mm) 0 125
Lamina, ply, laminae: unidirectional (UD) layer, is basic element, thicknesses (in mm) 0.125
(E‐Glass), 0.13 (Kevlar, HR carbon)
Skis, Ski Poles, Fishing Rods, Golf Clubs, Tennis and Badminton Rackets, Hockey Sticks, Poles(Pole vault),
Bicycle Frames, etc.
Laminate: a stack of two or more UD‐laminae with equal or different orientations
Laminate: a stack of two or more
with equal or different orientations
(„multiaxial“), defined by a stacking sequence
5
6
COMPOSITE MATERIALS
COMPOSITE MATERIALS
introduction
7
introduction
8
COMPOSITE MATERIALS
COMPOSITE MATERIALS
introduction
introduction
Advantages 2:
Advantages 1:
9Superb mechanical properties from high specific strength and high specific stiffness
9Superb mechanical properties from high specific strength and high specific stiffness,
(vehicles, sports goods, lightweight structures like satellites etc).
9Electrical conductivity: can vary from very low to intermediate levels, by control of fibers
and fillers. New: usage of C‐fibers as very long strain gauges
g
, g
p
p
gp
9Cost savings for fuel in vehicles, higher production speed for tools or rotating parts
9Non‐magnetic: useful as material for mine sweeper hulls, compass casings...
9Composites are „functional materials“, i.e. their anisotropy can be used for the tailoring of
p p
p
(
)
different properties, like the stiffness, or the coefficient of thermal expansion (CTE)
9Energy absorption: generally not as good as ductile metals, but good with special design
for crash absorbing under floor structures in helicopters, aircraft, cars
9Good corrosion resistance compared to metals, usage for tanks and storage containers,
valves, pipework, marine structures etc. (but not for Carbon/ Aluminium)
9Transparency to radiations, like x‐rays, radio waves. Application: radomes in A/C
9Fatigue: excellent compared to most metals, good damage tolerance, fewer inspections
required, benign failure modes
9Complex shapes: easier to fabricate than with sheet metals, applications: airframes for
helicopters, stealthy aircraft etc.
9Short‐run compatibility: composites can be produced for small quantity with cheaper
tooling than metals
9Dimensional stability: zero CTE design is possible, required for optical benches, precision
camera casings, satellite panels etc.
9Thermal conductivity: is very low through the thickness, can be high in fiber direction
9
10
COMPOSITE MATERIALS
MECHANICS OF COMPOSITES
introduction
overview
Disadvantages:
Composite material = heterogeneous and anisotropic materials
9Poor mechanical properties transverse to the fibers
Complex failure behavior
failure behavior of laminates
of laminates
9Complex
9Design process much more complex than with metals
9Repair methods more difficult/ or expensive
9Expensive materials and high production tools cost
Optical microscopy picture of CFRP
Optical microscopy picture of CFRP
plate with layup [0/90/0/90/0]
9Not very often economic necessity to apply composites, esp. in mechanical industry
9Lacking know‐how especially in the (mechanical) industry
• Macroscopic properties depend on the
microstructure
• Non‐uniform stress distribution
9Sustainability: energy‐intensive materials production and curing, problematic recycling of
thermoset matrices
h
i
11
12
MECHANICS OF COMPOSITES
DIFFERENT PROBLEMS AND APPLICATION
overview
Statement:
introduction
Geometry
MICRO
((constituents))
MACRO
((laminate))
MESO (ply)
MESO (ply)
What is distribution of stress?
What is the maximum value for stress?
What is the maximum value for stress?
Material
Constitutive behavior
[σ]
[ε]
How the structure
will be deformed?
13
matrix
Strength
Given: dimension of structure
Given:
dimension of structure
Find: what is maximum force?
or inverse
Gi
Given: forces
f
Find: what are dimensions?
Determine the constitutive behavior and strength properties to
calculate the structure response
fiber
d=3‐200 m ‐6
Basic components
Basic components
Material properties, voids,
distribution
14
MECHANICS OF COMPOSITES
MICROMECHANICS
h=0,1mm
Heterogeneous
Heterogeneous
continuum
Homogeneous, anisotropic
Homogeneous
anisotropic
continuum
Elastic and strength
properties of ply
Constitutive behavior
Ply theory
COMPOSITE LAMINATE
THEORY
MECHANICS OF COMPOSITES
objectives
course content
course content
You will learn HOW TO:
1 Reminding of continuum mechanics
1.
Reminding of continuum mechanics
1. determine the mechanical properties of an
unidirectional composite (elastic moduli, strength) as
function of mechanical properties of its constituents
(fiber, matrix)
2. Calculation of unidirectional composites (simple ply)
2 1 Mi
2.1 Micromechanical analysis and calculation of elastic properties
h i l
l i
d l l ti
f l ti
ti
2.2 Micromechanical analysis and fracture properties calculation
2.3 Macroscopic analysis and definition of elastic properties
2.4 Fracture criteria
2. define a constitutive behavior an unidirectional ply
3. Laminate composite
3.1 Design
3.2 Composite Laminate Theory
y
3.3 Fracture analysis
3 use different failure criteria
3.
use different failure criteria
4. apply the composite laminate theory
5. identify specific mechanical properties of composite
depending on its stacking
15
4. Particular problems: thick composites and hydrothermal analysis
16
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
reminder
Mechanical behavior of composite is based on the Continuum mechanics theory
which describes the behavior of the homogeneous linear elastic anisotropic (in
which describes the behavior of the homogeneous, linear elastic, anisotropic (in
general case)
Simple stress states
Homogeneous : It means, that material properties is identical in any considered
H
It
th t
t i l
ti i id ti l i
id d
point of the body.
normal stress state tension and compression for which the loading is applied
normal stress state tension and compression for which the loading is applied
perpendicular to surface of the element
Linear elastic: The relation between strains and stresses is described by linear
function.
pure shear loading when shear loading is applied in‐plane of the element
h
l di
h
h
l di i
li d i l
f th l
t
There exist simple stress states
F
F
Material symmetries: anisotropic, orthotropic, transversal isotropic and isotropic
tension
Material is isotropic
p if material properties in all directions are identical. As
p p
opposite case to isotropic is anisotropic body.
17
If material has 3 plane of symmetry (perpendicular to the principle axes of
If material has 3 plane of symmetry (perpendicular to the principle axes of
material), in this case it is called orthotropic. The elastic properties are identical
on both sides of each of the planes of symmetry.
compression
S
Complex stress state
Complex stress state is a combination of simple ones (traction‐traction,
Complex stress state is a combination of simple ones (traction
traction, simple
simple
bending, complex bending, etc.).
18
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
Deformations and stresses: external forces applied on body generate stresses and
deformations
deformations
Shear stress: τ= T/A0
Normal stress: σ= F/A0
Deformation normal longitudinal:
εL= (L‐L0)/L0 = ΔL/L0
Deformation normal transversal:
εT= (D‐D
(D D0)/D0 = ΔD/D
ΔD/D0
19
Shear deformations:
γ = ΔV/L0
reminder
Mathematical preliminaries: Vector definitions (*)
Summation rule, products (*)
3D stress state(*)
In general case of the classical material behavior the stress state is characterized by the
stress tensor.
stress tensor.
From mathematical point of view a second rank tensor.
Assuming orthonormal coordinate system: Cartesian coordinates xi with unit vectors ei which
g
have to fulfill the following conditions:
σ = σij eˆ i eˆ j
20
eˆ i = 1
eˆ i ⋅ eˆ j = δij
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
⎡ σ11 σ12
⎢
σ ( M ) = ⎢σ 21 σ 22
⎢⎣ σ31 σ32
3D stress state
σ13 ⎤
σ 23 ⎥⎥
σ33 ⎥⎦
reminder
Plane stress state
If thickness of a structure is small compared to other dimensions (plates thin
If thickness of a structure is small compared to other dimensions (plates, thin
shells), the normal stresses are neglected in terms (σ33 = 0, σ13 = 0, σ23 = 0).
So, it is considered that the structure is in a plane stress state.
with σij = σji
[σi ] = [σ2 , σ3 , σ4 ]T
[σi ] = [σ1 , σ 2 , σ6 ]T
As for stress tensor can be represented in the vector form (also called as Kelvin‐
Voigt notation).
21
11
22
33
23,32
13,31
12,21
1
2
3
4
5
6
⎡ σ11 ⎤ ⎡ σ1 ⎤ ⎡ σ11 ⎤ ⎡ σ1 ⎤
⎢σ ⎥ ⎢σ ⎥ ⎢σ ⎥ ⎢σ ⎥
⎢ 22 ⎥ ⎢ 2 ⎥ ⎢ 22 ⎥ ⎢ 2 ⎥
σ
σ
σ
σ
[σi ] = ⎢⎢ 33 ⎥⎥ = ⎢⎢ 3 ⎥⎥ = ⎢⎢ 33 ⎥⎥ = ⎢⎢ 3 ⎥⎥
σ4
τ
σ 23
τ 23
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 4⎥
⎢ σ13 ⎥ ⎢σ 5 ⎥ ⎢ τ13 ⎥ ⎢ τ5 ⎥
⎢⎣ σ12 ⎦⎥ ⎢⎣σ 6 ⎥⎦ ⎢⎣ τ12 ⎥⎦ ⎣⎢ τ6 ⎥⎦
22
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
reminder
Reciprocally,
Mathematical preliminaries: transformation rules (*)
T
Transformation rules for stress tensor by rotation about axis on angle θ.
f
ti
l f t
t
b
t ti
b t i
l θ
[σ i ](1, 2,3) = [Tσ ][σ i ]( x , y , z )
Stress tensor in
rotated coordinate
system (1,2,3)
23
2
s2
⎡ σ1 ⎤ ⎡ c
⎢σ ⎥ ⎢ 2
c2
⎢ 2⎥ ⎢ s
⎢ σ3 ⎥ ⎢ 0
0
⎢ ⎥=⎢
σ
0
0
⎢ 4⎥ ⎢
⎢
⎢ σ5 ⎥
0
0
⎢ ⎥ ⎢
⎣ σ6 ⎦ ⎣⎢ − sc sc
transformation
matrix
0 0
0 0
1 0
0 c
0 s
0 0
[Tσ]‐11 inverse of the transformation matrix [T
inverse of the transformation matrix [Tσ] , obtained by replacing
] obtained by replacing θ by ‐
by θ,
then transformation matrix for rotation of ‐θ written as:
Stress tensor in
reference coordinate
reference
coordinate
system (x,y,z)
2 sc ⎤ ⎡ σ xx ⎤
⎥ ⎢ ⎥
0
−2 sc ⎥ ⎢ σ yy ⎥
0
0 ⎥ ⎢ σ zz ⎥
⎥ ⎢ ⎥
0 ⎥ ⎢ σ yz ⎥
−s
c
0 ⎥⎥ ⎢ σ xz ⎥
⎢ ⎥ c = cosθ
0 c 2 − s 2 ⎦⎥ ⎣⎢ σ xy ⎦⎥ s = sin θ
[σ i ]( x, y , z ) = [Tσ ]−1 [σ i ](1, 2,3)
0
24
⎡c 2 s 2
⎢ 2
c2
⎢s
⎢0
0
[Tσ ]−1 = ⎢
0
⎢0
⎢0
0
⎢
⎢⎣ sc − sc
0
0
0
0
0
0
1
0
0
0
c
s
0 −s c
0
0
0
−2 sc ⎤
⎥
2 sc ⎥
0 ⎥
⎥
0 ⎥
0 ⎥⎥
c 2 − s 2 ⎥⎦
c = cosθ
s = sin θ
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
Displacement of a point
reminder
Deformation theory(*)
Initial state (or reference configuration)
before application of external forces
Final state (current configuration)
after application of external forces
3
Components of the strain tensor defined as:
1 ⎛ ∂u ∂u ∂u ∂u ⎞
εij = ⎜ i + j + k k ⎟
2 ⎝ ∂x j ∂xi ∂xi ∂x j ⎠
3
M
M
O
O
2
2
deformation
of the body
1
M’
It will be used the assumption of the small deformations, the second derivatives
are infinitively small compare to first order and can be neglected, so the
deformations are rewritten as follows:
[F]
1
1 ⎛ ∂u ∂u ⎞
εij = ⎜ i + j ⎟
2 ⎝ ∂x j ∂xi ⎠
During deformation of the solid body subjected to external forces, the point M
displaces to M’, the vector MM’ corresponds to the displacement of that point
u ( M ) = MM ' = ui ei
25
26
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
reminder
Strain (or deformation) tensor ε(M) is second rank
symmetric tensor:
symmetric tensor:
⎡ ε11 2ε12 2ε13 ⎤
ε ( M ) = ⎢⎢ 2ε 21 ε 22 2ε 23 ⎥⎥ =
⎣⎢ 2ε31 2ε32 ε33 ⎦⎥
⎡ ε11
⎢
= ⎢ 2ε12
⎣⎢ 2ε13
27
2ε12
ε 22
2ε 23
2ε13 ⎤ ⎡ ε11
2ε 23 ⎥⎥ = ⎢⎢ γ12
ε33 ⎦⎥ ⎣⎢ γ13
⎡ ε1 ⎤ ⎡ ε11 ⎤ ⎡ ε11 ⎤
⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎢ ε 2 ⎥ ⎢ ε 22 ⎥ ⎢ ε 22 ⎥
⎢ ε3 ⎥ ⎢ ε33 ⎥ ⎢ ε33 ⎥
⎢ ⎥=⎢
⎥=⎢ ⎥
⎢ ε 4 ⎥ ⎢ 2ε 23 ⎥ ⎢ γ 23 ⎥
⎢ ε5 ⎥ ⎢ 2ε13 ⎥ ⎢ γ13 ⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎢⎣ ε6 ⎥⎦ ⎢⎣ 2ε12 ⎥⎦ ⎢⎣ γ12 ⎥⎦
γ12
Transformation rules: changing of the coordinates system by rotating around axis
⎡ ε xx ⎤
⎡ ε xx ⎤
⎡ ε1 ⎤
⎢ε ⎥
⎢ε ⎥
⎢ε ⎥
⎢ yy ⎥
⎢ yy ⎥
⎢ 2⎥
⎢ ε zz ⎥
⎢ ε3 ⎥
-1 ⎢ ε zz ⎥
⎢ ⎥ = [Tε ] ⎢ ⎥ = [ R ][Tσ ][ R ] ⎢ ⎥
⎢ γ yz ⎥
⎢ γ yz ⎥
⎢ε4 ⎥
⎢ γ xz ⎥
⎢ γ xz ⎥
⎢ ε5 ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣⎢ ε6 ⎦⎥
⎣⎢ γ xy ⎦⎥
⎣⎢ γ xy ⎦⎥
γ13 ⎤
γ 23 ⎥⎥
ε33 ⎦⎥
ε 22
γ 23
A f
As for stress tensor, to simplify the constitutive law the
i lif h
i i l
h
strain tensor can be represented in the vector form
(also called as Kelvin‐Voigt notation).
11
22
33
23,32 13,31 12,21
1
2
3
4
5
= [ R ][Tσ ][ R ] [ ε i ]( x , y , z )
[εi ](1,2,3
1 2 3)
−1
deformation in the
rotated coordinate
d
di
system (1,2,3)
6
By replacing the components 2ε12, 2ε13, 2ε23 by strains γ12,
γ13, γ23 .
28
deformation in the
reference coordinate
f
i
transformation
system (x,y,z)
matrix
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
Generalized Hooke’s law (*):
reminder
General case of the anisotropic body:
absolute or invariant notation
b l
i
i
i
index notation
matrix form representation
σ = C :ε
ε = S :σ
σij = Cijkl ε kl
εij = Sijkl σkl
⎡ ε1 ⎤ ⎡ S11
⎢ε ⎥ ⎢ S
⎢ 2 ⎥ ⎢ 12
⎢ ε3 ⎥ ⎢ S13
⎢ ⎥=⎢
⎢ ε 4 ⎥ ⎢ S14
⎢ ε5 ⎥ ⎢ S15
⎢ ⎥ ⎢
⎣⎢ ε6 ⎦⎥ ⎣⎢ S16
ε j = Sij σi
σi = Cij ε j
{σ} = [C ]{ε}
{ε} = [ S ]{σ}
stiffness matrix
compliance matrix
Components Cij and Sij correspond to the material and depend on its elastic
properties (Young’ss moduli, Poisson
properties (Young
moduli, Poisson’ss ratios, Shear moduli)
ratios, Shear moduli)
⎡ σ1 ⎤ ⎡ C11
⎢ σ ⎥ ⎢C
⎢ 2 ⎥ ⎢ 12
⎢ σ3 ⎥ ⎢C13
⎢ ⎥=⎢
⎢σ4 ⎥ ⎢C14
⎢ σ5 ⎥ ⎢C15
⎢ ⎥ ⎢
⎢⎣ σ6 ⎥⎦ ⎢⎣C16
[C ] = [ S ]
−1
29
S12
S13
S14
S15
S 22
S 23
S 23
S33
S 24
S34
S 25
S35
S 24
S34
S 44
S 45
S 25
S35
S 45
S55
S 26
S36
S 46
S56
C12
C22
C13
C23
C14
C24
C15
C25
C23
C24
C33
C34
C34
C44
C35
C45
C25
C26
C35
C36
C45
C46
C55
C56
S16 ⎤
S 26 ⎥⎥
S36 ⎥
⎥
S 46 ⎥
S56 ⎥
⎥
S66 ⎦⎥
⎡ σ1 ⎤
⎢σ ⎥
⎢ 2⎥
⎢ σ3 ⎥
⎢ ⎥
⎢σ4 ⎥
⎢ σ5 ⎥
⎢ ⎥
⎣⎢ σ6 ⎦⎥
C16 ⎤
C26 ⎥⎥
C36 ⎥
⎥
C46 ⎥
C56 ⎥
⎥
C66 ⎥⎦
⎡ ε1 ⎤
⎢ε ⎥
⎢ 2⎥
⎢ ε3 ⎥
⎢ ⎥
⎢ε4 ⎥
⎢ ε5 ⎥
⎢ ⎥
⎢⎣ ε6 ⎥⎦
[C ] = [ S ]
−1
Ö 21 independent
material constants
enough to define
enough to define
compliance and stiffness
matrices
30
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
Orthotropic material (which has 3 plane of the
symmetry) suitable for the textile composites 2D
symmetry) suitable for the textile composites 2D
⎡ ε 1 ⎤ ⎡ S11
⎢ε ⎥ ⎢ S
⎢ 2 ⎥ ⎢ 12
⎢ε 3 ⎥ ⎢ S13
⎢ ⎥=⎢
⎢ε 4 ⎥ ⎢ 0
⎢ε 5 ⎥ ⎢ 0
⎢ ⎥ ⎢
⎣⎢ε 6 ⎦⎥ ⎣⎢ 0
S12
S13
0
0
S 22
S 23
S 23
S 33
0
0
0
0
0
0
S 44
0
0
0
0
S55
0
0
0
0
⎡σ 1 ⎤ ⎡C11 C12
⎢σ ⎥ ⎢C
⎢ 2 ⎥ ⎢ 12 C22
⎢σ 3 ⎥ ⎢C13 C23
⎢ ⎥=⎢
0
⎢σ 4 ⎥ ⎢ 0
⎢σ 5 ⎥ ⎢ 0
0
⎢ ⎥ ⎢
0
⎣⎢σ 6 ⎦⎥ ⎢⎣ 0
31
C13
C23
C33
0
0
0
0
0
0
C44
0
0
0
0
0
0
C55
0
0 ⎤ ⎡σ 1 ⎤
0 ⎥⎥ ⎢⎢σ 2 ⎥⎥
0 ⎥ ⎢σ 3 ⎥
⎥⎢ ⎥
0 ⎥ ⎢σ 4 ⎥
0 ⎥ ⎢σ 5 ⎥
⎥⎢ ⎥
S 66 ⎦⎥ ⎣⎢σ 6 ⎦⎥
0 ⎤ ⎡ε1 ⎤
0 ⎥⎥ ⎢⎢ε 2 ⎥⎥
0 ⎥ ⎢ε 3 ⎥
⎥⎢ ⎥
0 ⎥ ⎢ε 4 ⎥
0 ⎥ ⎢ε 5 ⎥
⎥⎢ ⎥
C66 ⎦⎥ ⎢⎣ε 6 ⎦⎥
reminder
Generalized Hooke’s law for the orthotropic material:
⎡ 1
⎢ E
⎢ 1
⎢ ν 21
−
⎡ ε1 ⎤ ⎢⎢ E2
⎢ε ⎥ ⎢
⎢ 2 ⎥ ⎢ − ν 31
⎢ ε3 ⎥
E3
⎢ ⎥ = ⎢⎢
ε
⎢ 4⎥ ⎢ 0
⎢ ε5 ⎥ ⎢
⎢ ⎥ ⎢
⎣⎢ ε 6 ⎦⎥ ⎢ 0
⎢
⎢
⎢ 0
⎣⎢
Ö 9 independent material
constants enough to
define compliance and
stiffness matrices
independent constants
dependent constants
32
ν12
E1
−
ν13
E1
0
0
1
E2
−
ν 23
E2
0
0
ν 32
E3
1
E3
0
0
0
0
1
G23
0
0
0
0
1
G13
0
0
0
0
−
−
⎤
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
1 ⎥
⎥
G12 ⎦⎥
⎡ σ1 ⎤
⎢ ⎥
⎢σ2 ⎥
⎢ σ3 ⎥
⎢ ⎥
⎢σ4 ⎥
⎢ σ5 ⎥
⎢ ⎥
⎣⎢ σ6 ⎦⎥
Sij = SSji Ö Ei = E j
υij υ ji
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
If the transversal plane of the symmetry can be defined then
constitutive relation for transversely isotropic material can be
constitutive relation for transversely isotropic material can be
used (suitable for the UD composite ply):
Definition of the stiffness matrix components for
orthotropic material:
orthotropic material:
C11 =
C22 =
C33 =
(1 − ν23 ν32 )E1
α
(1 − ν13 ν31 )E2
α
(1 − ν12 ν21 ) E3
C12 =
( ν21 + ν31ν23 ) E1 = ( ν12 + ν32 ν13 )E2
C13 =
( ν31 + ν21ν32 ) E1 = ( ν13 + ν12 ν23 ) E3
C23 =
α
C44 = G23
α
reminder
⎡ ε1 ⎤ ⎡ S11
⎢ε ⎥ ⎢ S
⎢ 2 ⎥ ⎢ 12
⎢ε 3 ⎥ ⎢ S12
⎢ ⎥=⎢
⎢ε 4 ⎥ ⎢ 0
⎢ε 5 ⎥ ⎢ 0
⎢ ⎥ ⎢
⎣⎢ε 6 ⎦⎥ ⎢⎣ 0
α
α
α
( ν32 + ν12 ν31 ) E2 = ( ν23 + ν21ν13 )E3
α
C55 = G13
S12
S 22
S 23
0
0
S12
S 23
S 22
0
0
0
0
0
0
C12
C22
C12
C23
0
0
0
0
C23
C22
0
0
0
0
0
0
C22 − C23
2
0
C66
0
0
0
0
⎡σ 1 ⎤ ⎡C11
⎢σ ⎥ ⎢C12
⎢ 2⎥ ⎢
⎢σ 3 ⎥ ⎢C12
⎢ ⎥=⎢
⎢σ 4 ⎥ ⎢ 0
⎢σ 5 ⎥ ⎢ 0
⎢ ⎥ ⎢
⎣⎢σ 6 ⎥⎦ ⎢⎣ 0
α
C66 = G12
with α = 1 − ν12 ν21 − ν13ν31 − ν23ν32 − 2 ν21ν32 ν13
33
0
0
0
0
0
0
2(S 22 − S 23 ) 0
0
S 66
0
⎤ ⎡σ 1 ⎤
⎥ ⎢σ ⎥
⎥ ⎢ 2⎥
⎥ ⎢σ 3 ⎥
⎥⎢ ⎥
⎥ ⎢σ 4 ⎥
⎥ ⎢σ 5 ⎥
⎥⎢ ⎥
S 66 ⎦⎥ ⎢⎣σ 6 ⎥⎦
0
0
0
0
0
Ö 5 independent material
constants enough to
define compliance and
stiffness matrices
0 ⎤ ⎡ε ⎤
1
0 ⎥⎥ ⎢ε ⎥
⎢ 2⎥
0 ⎥ ⎢ε ⎥
⎥ ⎢ 3⎥
0 ⎥ ⎢ε 4 ⎥
⎥
0 ⎥ ⎢ε 5 ⎥
⎢ ⎥
C66 ⎦⎥ ⎢⎣ε 6 ⎦⎥
i d
independent constants
d t
t t
dependent constants
effect of the transversal
ff
f h
l
plane of symmetry
34
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
Compliance matrix components for transversely isotropic material:
Components are identical in the transverse plane (2 3)
Components are identical in the transverse plane (2,3)
E1 = El
G23 = G32 = Gt
ν12 = ν13 = νlt
ν21 = ν31 = νtl
35
1
El
Stiffness matrix components for transversely isotropic material:
C11 =
E2 = E3 = Et
G12 = G13 = Glt
S11 =
reminder
ν23 = ν32 = νt
S12 = −
ν lt
El
=−
S 23 = −
1
S 66 =
Glt
2(S 22 − S 23 ) =
β
C22 = C33 =
ν tl
C44 =
Et
Et
1
Gt
2
t
Et
Gt =
2(1 + ν t )
νt
1
S 22 =
Et
(1−ν )E
C12 = C13 =
l
(1 − ν lt ν tl )Et
β
(C22 − C23 ) =
2
ν tl (1 + ν t )El ν lt (1 + ν t )Et
=
β
β
C23 =
Et
= Gt
2(1 + ν t )
(ν t + ν lt ν tl )Et
β
C55 = C
C66 = G
Glt
β = (1 + ν t )(1 − 2ν lt ν tl − ν t )
36
CONTINUUM MECHANICS
CONTINUUM MECHANICS
reminder
reminder
Compliance matrix components for isotropic material:
Generalized Hooke’s law for isotropic material:
⎡ ε1 ⎤ ⎡ S11
⎢ε ⎥ ⎢ S
⎢ 2 ⎥ ⎢ 12
⎢ε 3 ⎥ ⎢ S12
⎢ ⎥=⎢
⎢ε 4 ⎥ ⎢ 0
⎢ε 5 ⎥ ⎢ 0
⎢ ⎥ ⎢
⎢⎣ε 6 ⎥⎦ ⎢⎣ 0
S12
S12
0
0
S11
S12
0
0
S12
S11
0
0
0
0
0
0
0
0
0
0
⎡ C11 C12 C12
⎢
⎡ σ1 ⎤ ⎢C12 C11 C12
⎢σ ⎥ ⎢C12 C12 C11
⎢ 2⎥ ⎢
⎢ σ3 ⎥ ⎢ 0
0
0
⎢ ⎥=⎢
⎢σ4 ⎥ ⎢
⎢ σ5 ⎥ ⎢ 0
0
0
⎢ ⎥ ⎢
⎣⎢ σ6 ⎦⎥ ⎢
0
0
⎢ 0
⎣
2(S11 − S12 )
0
0
2(S11 − S12 )
0
0
0
0
0
1
C11 − C12
2
0
0
0
1
C11 − C12
2
0
0
⎤ ⎡σ 1 ⎤
⎥ ⎢σ ⎥
0
⎥ ⎢ 2⎥
⎥ ⎢σ 3 ⎥
0
⎥⎢ ⎥
0
⎥ ⎢σ 4 ⎥
⎥ ⎢σ 5 ⎥
0
⎥⎢ ⎥
2(S11 − S12 )⎥⎦ ⎢⎣σ 6 ⎥⎦
S11 =
0
⎤
⎥
⎥
⎥
0
⎥
⎥
0
⎥
⎥
⎥
0
⎥
⎥
1
C11 − C12 ⎥
2
⎦
0
0
⎡ ε1 ⎤
⎢ε ⎥
⎢ 2⎥
⎢ ε3 ⎥
⎢ ⎥
⎢ε 4 ⎥
⎢ ε5 ⎥
⎢ ⎥
⎣⎢ ε6 ⎦⎥
1
E
2(S11 − S12 ) =
Ö 2 independent material
constants enough to
define compliance and
define compliance and
stiffness matrices
S12 = −
ν
E
1
G
Stiffness matrix components for isotropic material:
E (1 − ν )
νE
C12 =
(1 + ν )(1 − 2ν )
(1 + ν )(1 − 2ν )
C11 − C12
E
=G
=
2
2(1 + ν )
C11 =
independent constants
dependent constants
37
38
Negative Poisson’s ratio video (*)
auxetics material (some minerals, paper… etc.)
MECHANICS OF COMPOSITES
UD COMPOSITES
course content
course content
micromechanics
Determination of the elastic properties of a composite which depend on:
¾ mechanical properties of its constituents (matrix and reinforcement)
mechanical properties of its constituents (matrix and reinforcement)
¾ redistribution of constituents
1 Reminding of continuum mechanics
1.
Reminding of continuum mechanics
2. Calculation of unidirectional composites (simple ply)
2 1 Mi
2.1 Micromechanical analysis and calculation of elastic properties
h i l
l i
d l l ti
f l ti
ti
2.2 Micromechanical analysis and fracture properties calculation
2.3 Macroscopic analysis and definition of elastic properties
2.4 Fracture criteria
Given:
1. properties of constituents: Ef EmGf Gm νf νm
2. their fractions: Vf Vm or Mf Mm
Find:
Engineering constants: E1E2G12ν12
⎡ σ1 ⎤ ⎡Q11
⎢σ ⎥ = ⎢Q
⎢ 2 ⎥ ⎢ 12
⎢⎣σ 6 ⎦⎥ ⎣⎢ 0
4. Particular problems: thick composites and hydrothermal analysis
39
ε = Sσ
σ = Qε
3. Laminate composite
3.1 Design
3.2 Composite Laminate Theory
y
3.3 Fracture analysis
40
Q12
Q22
0
0 ⎤⎡ ε1 ⎤
0 ⎥ ⎢ε 2 ⎥
⎥⎢ ⎥
Q66 ⎥⎦⎢⎣ε 6 ⎥⎦
⎡ ε1 ⎤ ⎡ S11
⎢ε ⎥ = ⎢ S
⎢ 2 ⎥ ⎢ 12
⎣⎢ε 6 ⎥⎦ ⎣⎢ 0
S12
S 22
0
0 ⎤⎡ σ1 ⎤
0 ⎥ ⎢σ 2 ⎥
⎥⎢ ⎥
S 66 ⎥⎦⎢⎣σ 6 ⎦⎥
2 possible approaches (out of many see review):
2
possible approaches (out of many see review):
¾ strength of the materials (rule of mixtures)
¾ semi‐empirical (Puck’s law, Rabiot’s law)
UD COMPOSITES
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micromechanics
micromechanics
Volume fraction:
Determination of the mass fraction:
Vi = v
= vi / v
/ vc
NF EN ISO 1172: Determination of the volume fraction by calcification for the glass
reinforced composites
vi volume of the constituent i, vc total volume of composite,
Vc = Vf + Vm + Vp = 1
Mf =
volume of the porosity (<1%: good quality, 5%:poor quality)
volume
of the porosity (<1%: good quality 5%:poor quality)
Mass fraction:
Mi = mi / mc
mi volume of constituent I,
l
f
i
mc total volume of composite
l l
f
i
Vf × ρ f
V f × ρ f + Vm × ρm
41
Vf =
Vf =
M f / ρf
M f Mm
+
ρf
ρm
m1: initial mass of the crucible
m2: initial mass of the crucible + specimen
m3: initial mass of the crucible + specimen after
m3: initial mass of the crucible + specimen after
calcification
Thickness measurement (prepregs)
Mc = Mf + Mm = 1
Mf =
m3 − m1
*100
m2 − m1
ρf and ρm densities of
reinforcement and matrix
with
htheoretical *V f _ theoretical
h: thickness of composite
h: thickness of composite
hmesured
htheoretical =
42
G
10* ρ f *V f _ theoretical
UD COMPOSITES
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micromechanics
Micromechanics models (*)
R l f i
Rule of mixtures: to determine the E1 and ν12
d
i
h E1 d 12
matrix
fibre
G: weight (g/m
G:
weight (g/m²))
ρf : density (g/cm3)
micromechanics
Rule of mixtures: to determine the E2 and G12
P
P
ε f = ε m = ε1
P = σ1S
P = σ f S f + σm Sm
P = Pf + Pm
σ1 = E1ε1 =
ε f = σ2 / E f
P ε1 E f S f ε1 Em S m
=
+
= ε1 ( E f V f + EmVm )
S
S
S
E1 = E f V f + EmVm
43
σ f = σm = σ2
P = E f ε f S f + Em ε m S m
ε m = σ 2 / Em
ε 2 = V f ε f + Vm ε m
1 V f Vm
=
+
E2 E f Em
ν12 = ν f V f + ν mVm
44
ε2 = V f
σ2
σ
+ Vm 2
Ef
Em
V f Vm
1
=
+
G12 G f Gm
44
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micromechanics
micromechanics: failure
micromechanics: failure
Semi‐empirical approach:
Puck’s law
E2 =
(
Em 1 + 0,85 ⋅ V f2
(1 − V )
1 25
1,25
f
)
E
+Vf m
Ef
G12 = Gm
G f (1 + V f ) + GmVm
G f Vm + Gm (1 + V f )
Rabiot’s law (applicable for the glass reinforced composites)
E2 = β E ' + (1 − β ) E ''
β = 0,197
E ' = E f V f + EmVm
1 V f Vm
=
+
E '' E f Em
matrix dominated ductile failure
Tension failure fiber
02
Th
The same for G12 but with
f G12 b
i h β = 0,
45
46
matrix dominated brittle failure
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micromechanics: failure
micromechanics: failure
The prediction of strength
The prediction of strength
Experiments
parameters using micromechanics
is difficult task.
Should be performed standard
Should
be performed standard
experiments to determine
strength
Micromechanics
(Rule of mixtures,etc…)
Case 1: εrf > εrm
σrf
Initiation of failure driven by failure of the matrix material
σ′f'
If Vf f < V
If V
< V’f , the failure of the matrix automatically results in fiber
, the failure of the matrix automatically results in fiber
failure, tensile stress of a composite is defined by Eq.2.
In case when Vf f >V’f failure of matrix occurs before failure of
fiber, and tensile stress of a composite defined by Eq.1
σrf
Stress
Objective: to determine the strength in longitudinal direction (direction of fiber)
of a composite based on the strength properties of the constituents (fiber and
of a composite based on the strength properties of the constituents (fiber and
matrix) and their redistribution.
micromechanics: failure
micromechanics: failure
σrm
εrm εrf
Strain
2 possible cases for the longitudinal failure:
εrf > εrm
σ r1 = σ rff V f Eq. 1
εrf < εrm
σ r1 = σ' f V f + σ rm (1 − V f )
σrm
47
48
0
V´f
Eq. 2
1
Vf fibre volume fraction
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micromechanics: failure
micromechanics: failure
Case 2: εrf < εrm
Case 1: εrf > εrm
Rapture of a composite is driven by
the failure of fibers.
σrf
σ r1
q
r1 = σ rm (1 − V f ) Eq. 4
0
σrm
σ′m
V´f Vfcrit
Multi‐cracking in the matrix material
1 Vf
If Vf < V’f , then fiber failure occurs before matrix one, and
tensile stress of composite defined by Eq 4
tensile stress of composite defined by Eq.4.
Fibers are more brittle than matrix
(carbon/epoxy)
Otherwise, Vf > V’f failure of fiber automatically results in
matrix failure in this case tensile stress defined by Eq 3
matrix failure, in this case tensile stress defined by Eq.3.
Stress concentration of one fiber
brings total rapture of the composite
brings total rapture of the composite.
εrm
εrf
Case 2: εrf < εrm
Matrix is more brittle than fibers
(glass/polyester)
(glass/polyester)
σ r1 = σ rf V f + σ m' (1 − V f ) Eq. 3
There isn’t any reinforcement effect σ
rm
when fiber volume fraction lower
than Vfcrit
than V
49
micromechanics: failure
micromechanics: failure
σrf
50
UD COMPOSITES
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lamina behavior
lamina behavior
lamina behavior
lamina behavior
Generalized Hooke’s law for 2D:
Relations for a orthotropic material:
Q11 =
σ = Qε
⎡ σ1 ⎤ ⎡Q11
⎢σ ⎥ = ⎢Q
⎢ 2 ⎥ ⎢ 12
⎣⎢σ 6 ⎦⎥ ⎣⎢ 0
Q12
Q22
0
0 ⎤⎡ ε1 ⎤
0 ⎥ ⎢ε 2 ⎥
⎥⎢ ⎥
Q66 ⎦⎥⎢⎣ε 6 ⎥⎦
ε = Sσ
⎡ ε1 ⎤ ⎡ S11
⎢ε ⎥ = ⎢ S
⎢ 2 ⎥ ⎢ 12
⎣⎢ε 6 ⎥⎦ ⎣⎢ 0
S12
S 22
0
Q12 = −
0 ⎤⎡ σ1 ⎤
0 ⎥ ⎢σ 2 ⎥
⎥⎢ ⎥
S 66 ⎥⎦⎣⎢σ 6 ⎦⎥
Q22 =
S11
E2
E2
E
=
=
= 2 Q11
S11S 22 − S122 1 − ν12 ν 21 1 − E2 ν 2
E1
12
E1
ν12 E2
ν 21 E1
S12
=
= ν12Q22 =
= ν 21Q11
1 − ν12 ν 21
S11S 22 − S122 1 − ν12 ν 21
Matrix form for a orthotropic material:
[Qij] reduced stiffness matrix
51
S 22
E1
E1
=
=
S11S 22 − S122 1 − ν12 ν 21 1 − E2 ν 2
12
E1
52
⎡ 1
⎢
⎡ ε1 ⎤ ⎢ E1
⎢ε ⎥ = ⎢− ν 21
⎢ 2 ⎥ ⎢ E2
⎣⎢ε 6 ⎦⎥ ⎢
⎢ 0
⎣⎢
ν12
E1
1
E2
−
0
⎤
0 ⎥
⎥ ⎡ σ1 ⎤
0 ⎥ ⎢σ 2 ⎥
⎥⎢ ⎥
σ
1 ⎥ ⎣⎢ 6 ⎥⎦
⎥
G12 ⎥⎦
Q66 =
1
= G12
S 66
UD COMPOSITES
UD COMPOSITES
tests
Determination of the mechanical properties
tests
Longitudinal:
Transverse:
In‐plane shear:
Torsion:
For complete elastic analysis of a orthotropic or transversely isotropic material we
l
l
l
f
h
l
l
need only 4 material properties defined:
E1 , E2 , G12 , ν12
¾ Tensile test (International Standard ISO 527‐4 for woven
and multiaxial composites and ISO 527‐5 for UD)
• tensile test in longitudinal direction
• tensile test in transverse direction
tensile test in transverse direction
E1 ,ν 12
Tensile test video
Gl fib (*)
Glass fiber (*)
E2
¾ Shear test (International Standard ISO 14129)
• tensile test for ± 45 ° specimen
G12
53
54
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tests
lamina behavior
lamina behavior
Off‐axis stiffness and transformation rules (*):
There is relation between components of the reduced stiffness matrix Qij in
There is relation between components of the reduced stiffness matrix Q
in
principle coordinates and components of the reduced stiffness matrix Q´ij in
arbitrary coordinate system (or off‐axis stiffness). The relation can be obtained
z
3
b
by rotating coordinate by angle Θ
t ti
di t b
l Θ around the 3 axis.
d th 3 i
[Q′] = [T ] [Q ][ R ][T ][ R ]
−1
2
−1
y
θ
⎡c
s
2s c ⎤
⎢ 2
⎥
[T ] = ⎢ s c2 −2s c ⎥
⎢
2
2⎥
⎣⎢ − s c s c c − s ⎦⎥
2
55
56
⎡1
[R] = ⎢⎢0
⎣⎢0
2
0
1
0
0⎤
0⎥
⎥
2⎥⎦
x
1
UD COMPOSITES
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lamina behavior
lamina behavior
Simplified relations between Q’ij and Qij
U3
U2
1
Components of the reduced stiffness in arbitrary
coordinate system are:
coordinate system are:
Q´11 = Q11
cos4θ
+ Q22
sin4θ
Q´12 = (Q11 + Q22 ‐ 4Q66)
+ 2(Q12 + 2Q66)
sin2θ
cos2
θ+ Q12
sin2θ
(sin4θ
stiffness transformation
stiffness transformation
cos2θ
+
cos4θ)
Q´16 = (Q11 ‐ Q12 ‐ 2Q66) sinθ cos3θ + (Q12 ‐ Q22 + 2Q66) sin3θ cosθ
Q´22 = Q11 sin4θ + 2(Q12 + 2Q66) sin2θ cos2θ + Q22 cos4 θ
Q´26 = (Q11 ‐ Q12 ‐ 2Q66) sin3θ cosθ + (Q12 ‐ Q22 + 2Q66) sinθ cos3θ
U1 = 1/8 (3Q11+ 3Q22 + 2Q12 + 4Q66)
Q’11
U1
cos(2θ)
cos(4θ)
Q’22
Q
U1
‐cos(2θ)
cos(2θ)
cos(4θ)
U2 = 1/2 (Q
1/2 (Q11
11 ‐ Q22)
Q’12
U4
‐cos(4θ)
U3 = 1/8 (Q11+ Q22 ‐ 2Q12 ‐ 4Q66)
Q’66
U5
(4θ)
‐cos(4θ)
Q’16
1/2.sin(2θ)
sin(4θ)
Q’26
1/2.sin(2θ)
‐sin(4θ)
U4 = 1/8 (Q11+ Q22 + 6Q12 ‐ 4Q66)
U5 = 1/8 (Q
1/8 (Q11+ Q
Q22 ‐ 2Q12 + 4Q
4Q66)
Handy equations for isotropic composites design which is the most conservative one.
Q´66 = [Q11 + Q22 ‐ 2(Q12 + Q66)] sin2θ cos2θ + Q66 (sin4θ + cos4θ)
57
58
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lamina behavior
lamina behavior
lamina behavior
lamina behavior
Simplified relation between S’ij and Sij
Transformation rule for compliance matrix
components S´ij
components S
S´11 = S11 cos4θ + S22 sin4θ + (2S12 + S66) sin2θ cos2θ
S´12 = (S11 + S22 ‐ S66
) sin2θ
cos2θ
+ S12
(sin4θ
+ cos4θ)
S´16 = [2(S
[2(S11 ‐ S12) ‐
) S66] sinθ
] i θ cos3θ + [2(S
[2(S12 ‐ S22) + S
) S66] sin
] i 3θ cosθ
θ
S´22 = S11 sin4θ + (2S12 + S66) sin2θ cos2θ + S22 cos4 θ
S´26 = [2(S11 ‐ S12) ‐ S66
] sin3θ
cosθ + [2(S12 ‐ S22) + S66] sinθ
cos3θ
S´66 = 2[2(S11 + S22 ‐ 2S12) ‐ S66)] sin2θ cos2θ + S66 (sin4θ + cos4θ)
1
V2
V3
SS’11
V1
cos(2θ)
cos(4θ)
S’22
V1
‐cos(2θ)
cos(4θ)
SS’12
V4
‐cos(4θ)
cos(4θ)
S’66
V5
‐4cos(4θ)
V1 = 1/8 (3S11+ 3S22 + 2S12 + S66)
V2 = 1/2 (S11 ‐ S22)
S’16
sin(2θ)
i (2θ)
2 i (4θ)
2sin(4θ)
S’26
sin(2θ)
‐2sin(4θ)
V3 = 1/8 (S
1/8 (S11+ SS22 ‐ 2S12 ‐ S66)
V4 = 1/8 (S11+ S22 + 6S12 ‐ S66)
V5 = 1/2 (S11+ S22 ‐ 2S12 + S66)
59
60
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lamina behavior
lamina behavior
lamina behavior
lamina behavior
In the reference coordinates (x,y,z):
⎡ε xx ⎤ ⎡ S
⎢ ⎥ ⎢
⎢ε yy ⎥ = ⎢ S
⎢γ xy ⎥ ⎢ S
⎣ ⎦ ⎣
'
11
'
12
'
16
⎡ 1
⎢
E
⎡ε xx ⎤ ⎢ x
⎢ ⎥ ⎢ ν xy
⎢ε yy ⎥ = ⎢− E
x
⎢γ xy ⎥ ⎢
⎣ ⎦ ⎢ m1
⎢ E
⎣ x
61
'
12
'
22
'
26
S
S
S
−
'
16
'
26
'
66
S
S
S
ν yyx
Ey
1
Ey
m2
Ey
1 cos 4 θ sin 4 θ 1 ⎛ 1 2ν12 ⎞ 2
⎟ sin 2θ
=
+
+ ⎜⎜
−
Ex
E1
E2
4 ⎝ G12
E1 ⎟⎠
⎤ ⎡σ xx ⎤
⎥⎢ ⎥
⎥ ⎢σ yy ⎥
⎥ ⎢σ xy ⎥
⎦⎣ ⎦
1
sin 4 θ cos 4 θ 1 ⎛ 1 2ν12 ⎞ 2
⎟ sin 2θ
=
+
+ ⎜⎜
−
Ey
E1
E2
4 ⎝ G12
E1 ⎟⎠
1
1
1 2ν12 ⎛ 1
1 2ν12
1 ⎞
⎟ cos 2 2θ
=
+
+
−⎜ +
+
−
G xy E1 E2
E1 ⎜⎝ E1 E2
E1 G12 ⎟⎠
m1 ⎤
⎥
Ex ⎥
⎡σ xx ⎤
m2 ⎥ ⎢ ⎥
σ yy
E y ⎥⎥ ⎢ ⎥
⎢σ ⎥
1 ⎥ ⎣ xy ⎦
Gxy ⎥⎦
ν xyy
Ex
=
ν yyx
Ey
=
ν12 ⎛ 1
1 2ν12
1 ⎞ sin 2 2θ
⎟
−
− ⎜⎜ +
+
E1 ⎝ E1 E2
E1 G12 ⎟⎠ 4
⎡ ⎛ 1 2ν 12 2 ⎞
⎛ 1 2ν 12 2 ⎞⎤
−
− ⎟⎟ − cs 3 ⎜⎜
−
− ⎟⎟⎥
m1 = E x ⎢c 3 s⎜⎜
E1
E1 ⎠
E1
E2 ⎠ ⎦
⎝ G12
⎣ ⎝ G12
Ex, Ey, Gxy, νxy and νyx (another notation E’1, E’2, G’12, ν’12 and ν’21) : elastic moduli
and Poisson’s coefficients in (x,y,z) coordinates.
(
)
m1 and m2 coupling coefficients shear/traction
⎡ ⎛ 1 2ν 12 2 ⎞ 3 ⎛ 1 2ν 12 2 ⎞⎤
−
− ⎟ − c s⎜⎜
−
− ⎟⎟⎥
m2 = E y ⎢cs 3 ⎜⎜
E1 E1 ⎟⎠
E1
E2 ⎠ ⎦
⎝ G12
⎣ ⎝ G12
UD COMPOSITES
laminate failure modes
laminate failure modes
63
Tension failure of cross‐ply CFRP
s = sin θ
62
UD COMPOSITES
Tension failure of multidirectional laminate
c = cos θ
failure criteria
failure criteria
The goal is to predict a failure of a lamina (or ply)
Assuming plane stress state
Assuming
plane stress state
Following criteria are considered:
¾ Maximum stress
¾ Maximum strain
¾ Tsai‐Hill
¾ Tsai‐Wu
¾ Hoffman
In‐plane shear failure of multidirectional laminate
64
UD COMPOSITES
UD COMPOSITES
failure criteria
failure criteria
failure criteria
failure criteria
Maximum stress criterion:
tension
σ1 / X+ < 1 (X+ : longitudinal tensile strength)
σ2 / Y+ < 1 (Y+ : transversely tensile strength)
compression
|σ1 / X‐ | < 1 (X‐
|σ2 / Y‐ | < 1 (Y‐
F11σ12 + 2 F12σ1σ 2 + F22σ 22 + F66σ62 + F1σ1 + F2σ 2 = 1
σ12
+
X +X −
: longitudinal compressive strength)
: transversely compressive strength)
tension
compression
65
shear
Tsai‐Hill
Tsai
Hill
+
X+ = X−
Y =Y
< 1 (V
( +: transversely tensile failure strain)
y
)
Hoffman
|ε1 / U‐ | < 1 (U‐: longitudinal compressive failure strain)
|ε2 / V
/ V‐ | < 1 (V
| < 1 (V‐: transversely compressive failure strain)
: transversely compressive failure strain)
|ε6 / G | < 1 (G: shear failure strain)
T iW
Tsai‐Wu
66
UD COMPOSITES
σ 22
σ62 ⎡ 1
1 ⎤
1 ⎤
⎡ 1
+
+ ⎢ + − − ⎥ σ1 + ⎢ + − − ⎥ σ 2 = 1
+ −
2
Y Y
S
X ⎦
Y ⎦
⎣X
⎣Y
−
X+ ≠ X−
Y+ ≠Y−
X+ ≠ X−
+
Y ≠Y
−
F12*
F12
−
−
1
2X 2
−0.014 ≤ −
1
+
2X X
−
−0.041 ≤ −
F12*
X + X −Y +Y −
1
2
Y
≤ −0.008
2X
Y +Y −
≤ −0.022
X +X −
−1 ≤ F12* ≤ 0
UD COMPOSITES
failure criteria
failure criteria
failure criteria
failure criteria
Tension of UD
67
X + X −Y +Y −
+
< 1 (U+: longitudinal tensile failure strain)
longitudinal
2 F12* σ1σ 2
Strengths
shear
|σ6 / S | < 1 (S : shear strength)
Maximum strain criterion
ε1 / U+
ε2 2 // V+
Fij σi σ j + Fi σi = 1
Quadratic criterion:
transversal
68
MECHANIC OF COMPOSITES
LAMINATES
design
design
course content
course content
1) each ply is denoted by a number representing the angle in degrees between
the main direction of the ply (direction 1 = fiber direction) and the x axis;
the main direction of the ply (direction 1 = fiber direction) and the x
2) adjacent plies are separated by slash (/) when their angles are different in
absolute value;
3) th
3) the sequence is described from the bottom face of the laminate z <0 to the
i d
ib d f
th b tt
f
f th l i t
0 t th
other side, brackets indicate the beginning and end of the stack;
4) if adjacent plies are identical (of the same orientation), then plies will be
represented by one number with superscript, which indicate the number of plies;
5) plies which are oriented at angles equal but opposite sign are denoted with ±;
6) superscript S placed after the bracket indicates a symmetric laminate (laminate
is said to be symmetric if the plies above the mid‐plane are a mirror image of
those below the mid‐plane);
7) symmetrical laminate with an odd number of plies is denoted identically to
7) symmetrical laminate with an odd number
of plies is denoted identically to
laminate having an even numbers of plies, ply of symmetry is underlined.
1 Reminding of continuum mechanics
1.
Reminding of continuum mechanics
2. Calculation of unidirectional composites (simple ply)
2 1 Mi
2.1 Micromechanical analysis and calculation of elastic properties
h i l
l i
d l l ti
f l ti
ti
2.2 Micromechanical analysis and fracture properties calculation
2.3 Macroscopic analysis and definition of elastic properties
2.4 Fracture criteria
3. Laminate composite
3.1 Design
3.2 Composite Laminate Theory
y
3.3 Fracture analysis
4. Particular problems: thick composites and hydrothermal analysis
[0/‐45/90/45/0]s
69
[0/‐45/902/60/0]
LAMINATES
LAMINATES
design
design
design
design
Cross ply or orthogonal consists
of many plies of 0° and 90
of many plies of 0
and 90°
degrees
Sign convention for describing directions
[0/‐45/90/45/0]s
[0/45/90/‐45/0]s
[[0/90]
/ ]s
71
[0/‐45/60]S
70
72
Angle ply or equilibrated [±θ]
consists of plies +θ° and ‐θ
consists of plies +θ
and θ °
LAMINATES
LAMINATES. THEORY OF PLATES
design
design
[0/‐45/90/60/30]
[0/‐45/902/60/0]
5 plies of the same
dimensions, consistent
of the same materials
of the same materials
but have 5 different
fiber orientations
6 plies,
the subscript 2 denotes two
plies of 90°
l
f °
A plate is a two‐dimensional structural element, i.e., one of the dimensions (the
plate thickness h) is small compared to the in‐plane dimensions a and b The load on
plate thickness h) is small compared to the in‐plane dimensions a and b. The load on
the plate is applied perpendicular to the center plane of the plate. In plate theory,
one generally distinguishes the following cases:
1.
2.
[0/‐45/60]S
[0/ 45/60]S
[0/‐45/60]
6 plies
3.
5 plies
Subscript s denotes the
symmetry of the stacking
from middle plane (again
the same orientation, ,
materials and dimensions)
Symmetry about the central
ply
73
Thick plates with a three‐dimensional stress state. These can only be described
by the full set of differential equations (rule of thumb: b/h < 5, a>b).
Thin plates with small deflections. The membrane stresses generated by the
deflection are small compared to the bending stresses and this simplifies the
analysis considerably. (rule of thumb: b/h > 5 and w< h/5).
Thin plates with large deflections. The
membrane stresses generated by the
d fl
deflection are significant compared to
f
d
the bending stresses and the plate
behaves nonlinearly. (rule of thumb:
b/h > 5 and w > h/5).
74
LAMINATES
LAMINATES
laminate theories
laminate theories
laminate theories
laminate theories
Macroscopic analysis based on the plate
theory.
There are some widely used theories:
• Love – Kirchhoff
• Reissner – Mindlin
• high‐order plate theories
•
•
75
76
J. N. Reddy. An Introduction to Continuum
Mechanics with Applications. Cambridge University
Press, New York, 2008.
J.N. Reddy Theory and Analysis of elastic plates and
Shells, 2nd ed., CRC Press, 207, p. 303
CPT ‐ Classical Plate Theory
FSDT ‐ First Order Shear Deformation Theory
TSDT ‐ Third Order Shear Deformation Theory
LAMINATES
LAMINATES
laminate theory
laminate theory
laminate theory
laminate theory
Assumptions:
o thin laminate composite plate is considered
thin laminate composite plate is considered
o the reference frame Oxyz defined in such way that
Oxy is middle‐plane of the laminate, Oz is
perpendicular; the upper and lower surfaces of
perpendicular; the upper and lower surfaces of
the laminate represent by z= ± h/2
o all assumption which correspond to the Love –
Ki hh ff l t th
Kirchhoff plate theory:
9 straight lines normal to the mid‐surface
remain straight after deformation
9 straight lines normal to the mid‐surface
remain normal to the mid‐surface after
deformation
9 the thickness of the plate does not change
during deformation.
o material is elastic, homogeneous
material is elastic, homogeneous
o small deformations
h0
hn
h1
hnn‐11 hk
h/2 x
h2
hk‐1
h/2
Ply k
z
For the ply k stress/strain relation in the O, x, y, z is written as:
[σ]k = [Qij' ]k [ε]
⎡σ xx ⎤
⎡Q'11
⎢σ ⎥ = ⎢Q'
⎢ yy ⎥
⎢ 12
⎣⎢σ xy ⎦⎥ k ⎢⎣Q'16
77
Q'12
Q' 22
Q' 26
Q'16 ⎤ ⎡ε xx ⎤
Q' 26 ⎥ ⎢⎢ε yy ⎥⎥
⎥
Q'66 ⎥⎦ k ⎢⎣ γ xy ⎦⎥
78
LAMINATES
LAMINATES
laminate theory
laminate theory
laminate theory
laminate theory
At any point of the laminate the deformations [ε] can be determined from
deformation of the plane [ε0] and the curatives [k] of the middle‐plane
Displacement field has 1
Displacement
field has 1st order (linearity along z)
order (linearity along z)
u ( x , y , z ) = u 0 ( x , y ) + zϕ x ( x , y )
⎡ε xx ⎤ ⎡ε 0xx ⎤ ⎡ κ x ⎤
⎡ε
⎢ε ⎥ = ⎢ε 0 ⎥ + z ⎢ κ ⎥
⎢ yy ⎥ ⎢ yy ⎥ ⎢ y ⎥
0
⎣⎢ γ xy ⎦⎥ ⎢⎣ γ xy ⎥⎦ ⎣⎢ κ xy ⎦⎥
v ( x , y , z ) = v0 ( x , y ) + z ϕ y ( x , y )
w( x , y , z) = w0 ( x , y)
For Classical Laminate theory, transversal shear deformations are negligible, so
displacement in‐plane than defined as:
membrane deformations
∂w ( x, y )
u ( x, y , z ) = u 0 ( x , y ) − z 0
∂x
v ( x, y , z ) = v 0 ( x, y ) − z
79
⎡ ∂u0 ⎤
⎢
⎥
∂x
⎡ε ⎤ ⎢
⎥
∂v0 ⎥
⎢ ⎥
ε m ( M ) = ⎢ε ⎥ = ⎢
⎢ ∂y
⎥
⎢γ ⎥ ⎢
⎣ ⎦
∂u0 ∂v0 ⎥
⎥
⎢
+
⎣⎢ ∂y ∂x ⎥⎦
0
xx
0
yy
0
xy
∂w0 ( x, y )
∂y
80
deformations correspond to
b
bending and torsion efforts
ff
⎡
∂ 2 w0 ⎤
⎢−z
⎥
∂x 2 ⎥
⎡ κx ⎤ ⎢
2
∂ w0 ⎥
⎢ ⎥
ε f ( M ) = z ⎢ κ y ⎥ = ⎢⎢ − z
∂y 2 ⎥
⎢ κ xy ⎥ ⎢
⎥
2
⎣ ⎦
⎢− 2 z ∂ w0 ⎥
∂x∂y ⎥⎦
⎣⎢
LAMINATES
LAMINATES
laminate theory
laminate theory
laminate theory
laminate theory
For ply k the stress/strain relation (in O, x, y, z) then written as:
⎡σ xx ⎤
⎡Q'11
⎢σ ⎥ = ⎢Q'
⎢ yy ⎥
⎢ 12
⎢⎣σ xy ⎥⎦ k ⎣⎢Q'16
Q'12
Q' 22
Q' 26
Q'16 ⎤ ⎡ε 0xx ⎤ ⎡Q'11
⎢ ⎥
Q' 26 ⎥ ⎢ε 0yy ⎥ + z ⎢Q'12
⎥
⎢
Q'66 ⎦⎥ k ⎢ε 0 ⎥ ⎣⎢Q'16
⎣ xy ⎦
⎡ N xx ⎤ h
h ⎡ σ xx ⎤
hk ⎡ σ xx ⎤
2
2
n
N ( x , y ) = ⎢⎢ N yy ⎥⎥ = ∫ σ(M )dz = ∫ ⎢⎢σ yy ⎥⎥ dz = ∑k =1 ∫ ⎢⎢σ yy ⎥⎥ dz
−h
−h
hk −1
2
2 ⎢σ ⎥
⎢⎣ N xy ⎥⎦
⎢⎣σ xy ⎥⎦ k
⎣ xy ⎦
Q'16 ⎤ ⎡ κ x ⎤
Q' 26 ⎥ ⎢⎢ κ y ⎥⎥
⎥
Q'66 ⎦⎥ k ⎣⎢ κ xy ⎥⎦
Q'12
Q' 22
Q' 26
⎡ Nx ⎤
n
N ( x , y ) = ⎢⎢ N y ⎥⎥ = ∑k =1 ∫ ([Q' ]k [ε 0 ] + z[Q' ]k [κ])dz
Resultant forces N(x,y) (expressed per unit length) obtained by integration
calculated stresses of each ply over the thickness:
z
⎣⎢ N xy ⎥⎦
hk
hk −1
[ ]
N ( x , y ) = [ A] ε 0 +[B ][κ ]
Nx
y
Nxy
Ny
Aij = ∑ k =1 (Q'ij )k (hk − hk −1 )
Ny
Nxy
n
Nxy
Nxy
x
81
Bij =
Nx
82
(
1 n
∑ (Q'ij )k hk2 − hk2−1
2 k =1
LAMINATES
)
LAMINATES
laminate theory
laminate theory
laminate theory
laminate theory
Resultant bending and torsion moments M(x,y) (expressed per unit length)
obtained by integration of the stresses of each ply over the thickness:
obtained by integration of the stresses of each ply over the thickness:
⎡Mx ⎤
hk
n
M ( x , y ) = ⎢⎢ M y ⎥⎥ = ∑k =1 ∫ [Q' ]k ε 0 z + [Q' ]k [κ]z 2 dz
d
hk −1
⎣⎢ M xy ⎦⎥
y
Mxy
Mxy
My
)
[ ]
M ( x , y ) = [B ] ε 0 +[D ][κ ]
My
Mxy
Mx
84
(
)
Bij =
1 n
∑k =1 (Q'ij )k hk2 − hk2−1
2
Dij =
1 n
∑ (Q'ij )k hk3 − hk3−1
3 k =1
x
83
[ ]
(
Mx
z
Mxy
⎡ M xx ⎤ h 2 ⎡σ xx ⎤
⎡M
hk ⎡ σ xx ⎤
n
⎢
⎥
⎢
⎥
M ( x , y ) = ⎢ M yy ⎥ = ∫ ⎢σ yy ⎥ z dz = ∑k =1 ∫ ⎢⎢σ yy ⎥⎥ z dz
h
hk −1
⎢⎣ M xy ⎥⎦ − 2 ⎢⎣σ xy ⎥⎦
⎢⎣σ xy ⎥⎦ k
(
)
LAMINATES
LAMINATES
laminate theory
laminate theory
laminate theory
laminate theory
Finally, resultant forces N(x,y) and moments M(x,y) as function of the
d f
deformations give constitutive behavior of the laminate:
ti
i
tit ti b h i
f th l i t
⎡ N x ⎤ ⎡ A11
⎢ N ⎥ ⎢A
⎢ y ⎥ ⎢ 12
⎢ N xy ⎥ ⎢ A16
⎢ M ⎥ = ⎢B
⎢ x ⎥ ⎢ 11
⎢ M y ⎥ ⎢ B12
⎢ M xy ⎥ ⎢ B
⎣
⎦ ⎣ 16
A12
A22
A26
B12
B22
B26
Aij = ∑ k =1 (Q'ij )k (hk − hk −1 )
n
85
A16
A26
A66
B16
B26
B66
B11
B12
B16
D11
D12
D16
B12
B22
B26
D12
D22
D26
B16 ⎤ ⎡ ε 0xx ⎤
⎢ ⎥
B26 ⎥ ⎢ ε 0yy ⎥
⎥ 0
B66 ⎥ ⎢ γ xy ⎥
D16 ⎥ ⎢ κ x ⎥
⎥⎢ ⎥
D26 ⎥ ⎢ κ y ⎥
D66 ⎥⎦ ⎢⎣ κ xy ⎥⎦
extensional stiffness matrix components
(
)
coupling stiffness matrix components
(
)
bending stiffness matrix components
Bij =
1 n
∑k =1 (Q'ij )k hk2 − hk2−1
2
Dij =
1 n
∑ (Q'ij )k hk3 − hk3−1
3 k =1
86
LAMINATES
LAMINATES
laminate theory
laminate theory
⎡ N x ⎤ ⎡ A11
⎢ N ⎥ ⎢A
⎢ y ⎥ ⎢ 12
⎢ N xy ⎥ ⎢ A16
⎢ M ⎥ = ⎢B
⎢ x ⎥ ⎢ 11
⎢ M y ⎥ ⎢ B12
⎥ ⎢
⎢
⎣ M xy ⎦ ⎣ B16
87
A12
A22
A26
B12
B22
B26
A16
A26
A66
B16
B26
B66
B11
B12
B16
D11
D12
D16
B12
B22
B26
D12
D22
D26
B16 ⎤ ⎡ ε 0xx ⎤
⎢ ⎥
B26 ⎥ ⎢ ε 0yy ⎥
⎥ 0
B66 ⎥ ⎢ γ xy ⎥
D16 ⎥ ⎢ κ x ⎥
⎥⎢ ⎥
D26 ⎥ ⎢ κ y ⎥
D66 ⎥⎦ ⎢⎣ κ xy ⎥⎦
•
•
•
•
laminate theory
laminate theory
⎡ N x ⎤ ⎡ A11
⎢N ⎥ ⎢
⎢ y ⎥ ⎢ A12
⎢ N xy ⎥ ⎢ 0
⎢
⎥=⎢
⎢ M x ⎥ ⎢ B11
⎢ M y ⎥ ⎢ B12
⎥ ⎢
⎢
⎢⎣ M xy ⎥⎦ ⎢⎣ B16
traction/shear coupling
t ti /b di
traction/bending coupling
li
traction/torsion coupling
bending/torsion coupling
88
A12
0
B11
B12
A22
0
0
A66
B12
B16
B22
B26
B12
B22
B16
B26
D11
D12
D12
D22
B26
B66
D16
D26
B16 ⎤ ⎡ ε 0xx ⎤
⎢ ⎥
B26 ⎥⎥ ⎢ ε 0yy ⎥
B66 ⎥ ⎢ γ 0xy ⎥
⎥⎢ ⎥
D16 ⎥ ⎢ κ x ⎥
D26 ⎥ ⎢ κ y ⎥
⎥⎢ ⎥
D66 ⎦⎥ ⎢⎣ κ xy ⎥⎦
•
•
•
•
no traction/shear coupling
t ti /b di
traction/bending coupling
li
traction/torsion coupling
bending/torsion coupling
LAMINATES
LAMINATES
laminate theory
laminate theory
laminate theory
laminate theory
Not equilibrated, symmetric
Orthogonal non symmetric
Orthogonal non symmetric
⎡ N x ⎤ ⎡ A11
⎢N ⎥ ⎢
⎢ y ⎥ ⎢ A12
⎢ N xy ⎥ ⎢ 0
⎢
⎥=⎢
⎢ M x ⎥ ⎢ B11
⎢ M y ⎥ ⎢ B12
⎢
⎥ ⎢
⎢⎣ M xy ⎦⎥ ⎣⎢ 0
A12
A22
0
0
B11
B12
B12
B22
0
B12
B22
A66
0
0
0
D11
D12
0
D12
D22
0
B66
0
0
0 ⎤ ⎡ ε 0xx ⎤
⎢ ⎥
0 ⎥⎥ ⎢ ε 0yy ⎥
B66 ⎥ ⎢ γ 0xy ⎥
⎥⎢ ⎥
0 ⎥⎢ κx ⎥
0 ⎥⎢ κ y ⎥
⎥⎢ ⎥
D66 ⎥⎦ ⎢⎣ κ xy ⎦⎥
•
•
•
•
⎡ N x ⎤ ⎡ A11
⎢N ⎥ ⎢
⎢ y ⎥ ⎢ A12
⎢ N xy ⎥ ⎢ A16
⎢
⎥=⎢
⎢Mx ⎥ ⎢ 0
⎢My ⎥ ⎢ 0
⎢
⎥ ⎢
⎢⎣ M xy ⎦⎥ ⎢⎣ 0
no traction/shear coupling
t ti /b di
traction/bending coupling
li
no traction/torsion coupling
no bending/torsion coupling
89
A12
A16
0
0
A22
A26
A26
A66
0
0
0
0
0
0
0
0
D11
D12
D12
D22
0
0
D16
D26
0 ⎤ ⎡ ε 0xx ⎤
⎢ ⎥
0 ⎥⎥ ⎢ ε 0yy ⎥
0 ⎥ ⎢ γ 0xy ⎥
⎥⎢ ⎥
D16 ⎥ ⎢ κ x ⎥
D26 ⎥ ⎢ κ y ⎥
⎥⎢ ⎥
D66 ⎥⎦ ⎢⎣ κ xy ⎥⎦
•
•
•
•
90
LAMINATES
LAMINATES
laminate theory
laminate theory
failure analysis
failure analysis
¾ Multi‐scale nature (micro/meso/macro)
¾ Different mechanisms:
Different mechanisms:
• fiber breakage
• matrix cracking
• fiber/matrix decohesion
fib /
i d h i
• delamination
¾ coupling
¾ internal
g
y
y
Orthogonal symmetry
⎡ N x ⎤ ⎡ A11
⎢N ⎥ ⎢
⎢ y ⎥ ⎢ A12
⎢ N xy ⎥ ⎢ 0
⎢
⎥=⎢
⎢Mx ⎥ ⎢ 0
⎢My ⎥ ⎢ 0
⎥ ⎢
⎢
⎢⎣ M xy ⎦⎥ ⎢⎣ 0
91
traction/shear coupling
no traction/bending coupling
t ti /b di
li
no traction/torsion coupling
bending/torsion coupling
A12
A22
0
0
0
0
0
0
0
0
0
A66
0
0
0
D11
D12
0
D12
D22
0
0
0
0
⎤ ⎡ ε xx ⎤
⎥ ⎢ ε0 ⎥
⎥ ⎢ yy ⎥
0 ⎥ ⎢ γ 0xy ⎥
⎥⎢ ⎥
0 ⎥⎢ κx ⎥
0 ⎥⎢ κ y ⎥
⎥⎢ ⎥
D66 ⎥⎦ ⎢⎣ κ xy ⎦⎥
0
0
0
•
•
•
•
no traction/shear coupling
no traction/bending coupling
t ti /b di
li
no traction/torsion coupling
no bending/torsion coupling
92
LAMINATES
LAMINATES
failure analysis
failure analysis
failure analysis
failure analysis
Damage evolution
Define the
configuration of the
fi
ti
f th
laminate and
boundary conditions
Calculation of the
deformation filed of the
laminate εxx, εyy, γxy
Determinations of
the constitutive
matrices [A], [B], [D]
93
Choose a failure
Choose
a failure
criterion for each
ply. Determination
of loading
of loading
First ply failure
(FPF)
93
Calculation deformations ε11,
ε22, γ12 and stresses σ11, σ22,
τ12 in material coordinates at
the bottom (z=hk) and top
( k‐1)
(z=h
) surfaces of each ply
Last ply failure
(LPF)
94
LAMINATES
LAMINATES
failure analysis
failure analysis
failure analysis
failure analysis
Determination of the last ply failure (LPF)
Determination of the last ply failure (LPF)
Complete ply failure (CPF)
Complete ply failure (CPF)
Partial ply failure (PPF)
Partial ply failure (PPF)
After determining the FPF, repeat the calculation to determine failure of the
second ply and so on until the last ply is broken
second ply, and so on until the last ply is broken.
Whatever the mode of failure, it
py
is assumed that the ply is no
longer able to support the load
However, for each iteration, we must take into account the plies where the
f t
fracture was determined. How? The failed ply remains physically in the same
d t
i d H ? Th f il d l
i
h i ll i th
place, but some elastic characteristics are reduced.
E2 = G12 = 0
E1 ≠ 0
Two approaches are generally adopted:
E1 = E2 = G12 = 0
Complete ply failure (CPF)
Partial ply failure (PPF)
95
1) Transversal or shear
damage
g
96
2) Fiber failure
E1 = E
= E2 = G
= G12 = 0
=0
MECHANIC OF COMPOSITES
THICK COMPOSITES
course content
laminate theory
a
ate t eo y
1 Reminding of continuum mechanics
1.
Reminding of continuum mechanics
2. Calculation of unidirectional composites (simple ply)
2 1 Mi
2.1 Micromechanical analysis and calculation of elastic properties
h i l
l i
d l l ti
f l ti
ti
2.2 Micromechanical analysis and fracture properties calculation
2.3 Macroscopic analysis and definition of elastic properties
2.4 Fracture criteria
Laminate theory with taking into account transversal shear
y
g
The classical laminate theory becomes quite unsuitable in the case of thick
composites (relation width/thickness is less than 10) to describe the mechanical
behavior An improvement consists of taking into account the transverse shear
behavior. An improvement consists of taking into account the transverse shear
displacement of the 1st order (linear along z).
3. Laminate composite
3.1 Design
3.2 Composite Laminate Theory
y
3.3 Fracture analysis
Th t i t
The strain tensor has 5 nonzero components:
h 5
t
4. Particular problems: thick composites and hydrothermal analysis
97
98
⎡ ε xx
⎢
ε ( M ) = ⎢ε xy
⎢ ε xz
⎣
THICK COMPOSITES
ε xy
ε yy
ε yz
ε xz ⎤
⎥
ε yz ⎥
0 ⎥⎦
THICK COMPOSITES
laminate theory
a
ate t eo y
laminate theory
a
ate t eo y
Deformation field consists of:
1) D f
1) Deformation field of membrane and flexion
ti fi ld f
b
d fl i
The normal to the mid‐surface remains straight but not necessarily perpendicular
to the mid‐surface
h
d
f
⎡ ∂ϕ x
⎤
⎡ ∂u0 ⎤
⎢
⎥
⎢
⎥
0
∂x
∂x
⎢
⎥
⎥
⎡ ε xx ⎤ ⎡ ε xx ⎤
⎡ κx ⎤ ⎢
⎢ ∂ϕ y
⎥
⎢ ⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ∂v0 ⎥
ε mf ( M ) = ⎢ε yy ⎥ = ⎢ε yy ⎥ + z ⎢ κ y ⎥ = ⎢
+ z⎢
⎥
⎥
∂y
∂y
⎢
⎥
⎥
⎢ γ xy ⎥ ⎢ γ 0 ⎥
⎢ κ xy ⎥ ⎢
⎣ ⎦ ⎢⎣ xy ⎥⎦
⎣ ⎦ ⎢ ∂u ∂v ⎥
⎢
⎥
∂ϕ
∂ϕ
y
0
0
⎢ x+
⎥
⎢ ∂y + ∂x ⎥
∂x ⎦
⎣
⎦
⎣ ∂y
According to the displacement field of the first order adopted as:
According to the displacement field of the first order adopted as:
2) transversal deformations
u ( x, y , z ) = u0 ( x, y ) + zϕ x ( x, y )
v ( x, y , z ) = v0 ( x, y ) + zϕ y ( x, y )
w ( x, y , z ) = w0 ( x, y )
99
w0
⎡ ∂∂w
⎤
+ ϕx ⎥
⎢
⎡ γ yz ⎤
x
∂
γ c (M ) = ⎢ ⎥ =⎢ ∂w
⎥
⎣ γ xz ⎦ ⎢ 0 + ϕ y ⎥
⎥⎦
⎢⎣ ∂y
100
THICK COMPOSITES
THICK COMPOSITES
laminate theory
a
ate t eo y
laminate theory
a
ate t eo y
Resultant forces in shear Q(x,y) for the laminate or multi‐layer composite are
obtained by integrating the stresses of each ply over the thickness of the
obtained by integrating the stresses of each ply over the thickness of the
composite:
Expression of the resultant membrane forces N(x,y), resultant moments M(x,y)
and resultant shear forces Q(x y) as function of the deformations of composite can
and resultant shear forces Q(x,y) as function of the deformations of composite can
be written in the matrix form as:
hk
2 σ
⎡Qy ⎤
⎡ yz ⎤
⎡σ ⎤
n
Q ( x, y ) = ⎢ ⎥ = ∫ ⎢ ⎥ ddz = ∑ k =1 ∫ ⎢ yz ⎥ ddz
hk −1 ⎣ σ xz ⎦
⎣Qx ⎦ − h 2 ⎣ σ xz ⎦
k
hk
⎡Q ⎤
n
Q ( x, y ) = ⎢ y ⎥ = ∑ k =1 ∫ [C ']k [ γ c ] dz
hk −1
⎣Qx ⎦
h
Q ( x, y ) = [ F ][ γ c ]
⎡ N x ⎤ ⎡ A11
⎢ N ⎥ ⎢A
⎢ y ⎥ ⎢ 12
⎢ N xy ⎥ ⎢ A16
⎢
⎥ ⎢
⎢ M x ⎥ = ⎢ B11
⎢ M y ⎥ ⎢ B12
⎢
⎥ ⎢
⎢ M xyy ⎥ ⎢ B16
⎢Q ⎥ ⎢ 0
⎢ y ⎥ ⎢
⎢⎣ Qx ⎦⎥ ⎢⎣ 0
z
Fij = ∑ k =1 ( C 'ij )k ( hk − hk −1 )
y
n
Qx
Qy
n
Qx
x
B11
B12
B16
D11
D12
D16
0
0
B12
B22
B26
D12
D22
D26
0
0
B16
B26
B66
D16
D26
D66
0
0
0
0
0
0
0
0
F44
F45
0 ⎤⎡ε 0xx ⎤
⎢ ⎥
0 ⎥⎥⎢ ε 0yy ⎥
0 ⎥⎢ γ 0xy ⎥
⎥⎢ ⎥
0 ⎥⎢ κ x ⎥
0 ⎥⎢ κ y ⎥
⎥⎢ ⎥
0 ⎥⎢ κ xyy ⎥
F45 ⎥⎢ γ 0yz ⎥
⎥⎢ ⎥
F55 ⎦⎥⎢⎣ γ 0xz ⎦⎥
′ = G23 cos 2 θ + G13 sin 2 θ
C44
′ = (G13 − G23 )sin θ cos θ
C45
′ = G13 cos 2 θ + G23 sin 2 θ
C55
102
THICK COMPOSITES
SANDWITCH MATERIALS
laminate theory
a
ate t eo y
Limitations of the laminate theory of thick composites
de t o
definition
Describes the behavior of a beam, plate, or shell which consists of three layers ‐
two facesheets (skins) and one core
two facesheets
(skins) and one core
According to improvement, it is not negligible the shear stresses in the skin
But for this approach, the transverse shear deformations are independent of z, so
f h
h h
h
d f
d
d
f
constant along thickness. This implies that the shear stresses are piecewise
constant in thickness, so discontinuous in the interfaces.
For a better description of the mechanical behavior of thick composites, it is
y
ff
f
necessary to introduce correction coefficients of shear:
⎡Q y ⎤ ⎡ H 44
⎢Q ⎥ = ⎢ H
⎣ x ⎦ ⎣ 45
H 45 ⎤
[γ c ]
H 55 ⎥⎦
H ij = k ij Fij
103
A16
A26
A66
B16
B26
B66
0
0
Fij = ∑k =1 (C 'ij )k (hk − hk −1 )
Qy
101
A12
A22
A26
B12
B22
B26
0
0
104
SANDWITCH MATERIALS
SANDWITCH MATERIALS
Hypothesis/Notations
ypot es s/ otat o s
o h is core thickness, h1
is core thickness, h1 and h2
and h2 are thicknesses
are thicknesses of the upper and lower skins
of the upper and lower skins
respectively
o the thickness of the core is much larger than those of skin
o the displacement of the core is linear function of z
the displacement of the core is linear function of z
o displacements of the facesheets are uniform in‐plane
o the core transmits the transverse shear stress only
o transverse shear stresses are negligible in the facesheets
t
h
t
li ibl i th f
h t
105
constitutive equations
constitutive equations
Expression of the resultant membrane forces N(x,y), resultant moments M(x,y)
and resultant shear forces Q(x y) as function of the deformations of sandwich can
and resultant shear forces Q(x,y) as function of the deformations of sandwich can
be written in the matrix form as:
⎡ N x ⎤ ⎡ A11 A12
⎢ N ⎥ ⎢A
⎢ y ⎥ ⎢ 12 A22
⎢ N xy ⎥ ⎢ A16 A26
⎢
⎥ ⎢
⎢ M x ⎥ = ⎢C11 C12
⎢ M y ⎥ ⎢C12 C22
⎢
⎥ ⎢
⎢ M xy ⎥ ⎢C16 C26
⎢Q ⎥ ⎢ 0
0
⎢ y ⎥ ⎢
0
⎣⎢ Qx ⎦⎥ ⎣⎢ 0
A16
A26
A66
C16
C26
B11
B12
B16
D11
D12
B12
B22
B26
D12
D22
B16
B26
B66
D16
D26
0
0
0
0
0
C66
0
0
D16
0
0
D26
0
0
D66
0
0
0
F44
F45
106
SANDWITCH MATERIALS
HYDROTHERMAL ANALYSIS
co st tut e equat o s
constitutive equations
general statement
general statement
The materials deform when the temperature changes from T0 to T or when they
absorb water or moisture.
b b t
it
In the linear range, assuming that the temperature and the concentration of water
is uniform in the volume, we have:
• p
properties dependent of facesheets
p
p
properties:
p
p
Aij = Aij1 + Aij2
Cij = Cij1 + Cij2
Aij1j = ∑ k =1 ( Q 'ijj )k ( hk − hk −1 )
h 2
( Aij − Aij1 )
2
h
Dij = ( Cij2 − Cij1 )
2
Bij =
n1
Cij1 =
• properties dependent of core properties:
1 n1
Q 'ij )k ( hk2 − hk2−1 )
∑
k =1 (
2
T
T
⎤ ⎡ε xxH ⎤ ⎡ε xxM ⎤ ⎡ε xx
⎤ ⎡ε xxH ⎤
⎡σ xx ⎤
⎡ε xx ⎤ ⎡ε xx
⎢ ⎥
⎢ ⎥ ⎢ T ⎥ ⎢ H⎥ ⎢ M⎥ ⎢ T ⎥ ⎢ H⎥
⎢ε yy ⎥ = ⎢ε yy ⎥ + ⎢ε yy ⎥ + ⎢ε yy ⎥ = ⎢ε yy ⎥ + ⎢ε yy ⎥ + [S ']⎢σ yy ⎥
T ⎥
⎢ H⎥ ⎢ M⎥ ⎢ T ⎥ ⎢ H⎥
⎢σ xy ⎥
⎢γ xy ⎥ ⎢γ xy
⎣ ⎦ ⎣ ⎦ ⎣γ xy ⎦ ⎣γ xy ⎦ ⎣γ xy ⎦ ⎣γ xy ⎦
⎣ ⎦
Fij = h ⋅ Cij′a
[ε ]
• in case of the symmetric sandwich (top and bottom facesheets are equal)
Aij1 = Aij2
Cij1 = −Cij2
107
⎤⎡ε xx0 ⎤
⎥⎢ 0 ⎥
⎥⎢ε yy ⎥
⎥⎢γ xy0 ⎥
⎥⎢ ⎥
⎥⎢ κ x ⎥
⎥⎢ κ y ⎥
⎥⎢ ⎥
0 ⎥⎢κ xy ⎥
F45 ⎥⎢γ yz0 ⎥
⎥⎢ ⎥
F55 ⎦⎥⎣⎢γ xz0 ⎦⎥
0
0
0
0
0
Aij = 2 Aij2
[ε ]
[ε ]
[ε ]
Dij = hC
h ij2
Bij = Cij = 0
108
total deformations
T
thermal deformations
thermal deformations
H
hydro deformations
M
mechanical deformations
HYDROTHERMAL ANALYSIS
HYDROTHERMAL ANALYSIS
UD
UD
⎡ ε 1 ⎤ ⎡ S11
⎢ε ⎥ = ⎢ S
⎢ 2 ⎥ ⎢ 12
⎢⎣ε 6 ⎥⎦ ⎢⎣ 0
S12
S 22
0
y
Off‐axis ply
In the material coordinates, the constitutive behavior is written as
,
2
0 ⎤ ⎡σ 1 ⎤ ⎡α 1 ⎤
⎡ β1 ⎤
⎥
⎢
⎥
⎢
⎥
0 ⎥ ⎢σ 2 ⎥ + ⎢α 2 ⎥ ΔT + ⎢⎢ β 2 ⎥⎥ Δc
⎢⎣ 0 ⎥⎦
S 66 ⎥⎦ ⎢⎣σ 6 ⎥⎦ ⎢⎣ 0 ⎥⎦
θ
⎡ ε1 ⎤ ⎡ S '11
⎢ε ⎥ = ⎢ S '
⎢ 2 ⎥ ⎢ 12
⎢⎣ε 6 ⎦⎥ ⎢⎣ S '16
with:
(α1, α2) coefficients of the thermal expansion in longitudinal and transversal
directions (µm m‐1.K
directions (µm.m
K‐1 or 10‐6 K‐1)
ΔT variation of the temperature
(β1, β2) swelling coefficients (m.m
swelling coefficients (m m‐11.kg. kg
kg kg‐11)
Δc mass fraction od absorbed water (kg. kg‐1)
109
110
REFERENCES
French:
• Matériaux composites : comportement mécanique et analyse des structures, Jean‐Marie BERTHELOT, TEC et DOC –
2005
• Calcul et conception des structures composites, Pierre ODRU, Techniques de l’Ingénieur, A 7 792.
• Comportement élastique et viscoélastique des composites, par Yvon CHEVALIER, A 7 750.
• Critères de rupture des composites – Approche macroscopique, Yvon CHEVALIER, Techniques de l’Ingénieur, A 7 755.
• Matériaux
M té i
composites,
it Daniel
D i l GAY,
GAY Hermès
H
è Science
S i
P bli ti
Publications
– 2005
English:
• Greenhalgh, E.S. Failure analysis and fractigraphy of polymer composites, Woodhead Publishing Series in Composites
Science and Engineering No. 27, 2009, 608 pages.
• Daniel,
D i l I.M.,
I M Ishai,
I h i O.:
O Engineering
E i
i Mechanics
M h i off Composite
C
it Materials,
M t i l Oxford
O f d Univ.
U i Press,
P
1994
• D. Gay and S.V. Hoa, "Composite materials, Design and application", CRC Press, second edition, ISBN: 978‐1‐4200‐
4519‐2, (2007).
• E. Barbero, "Finite element analysis of composite materials", CRC Press, ISBN:978‐1‐4200‐5433‐0, (2008).
• J.
J N.
N Reddy.
Reddy An Introduction to Continuum Mechanics with Applications.
Applications Cambridge University Press,
Press New York,
York
2008.
• J.N. Reddy Theory and Analysis of elastic plates and Shells, 2nd ed., CRC Press
Free calculation tool:
• eLamX: Laminate theory,
theory Java,
Java [ABD] matrix,
matrix 3D failure envelope plots,
plots http:/ /tu‐dresden.de.
/tu dresden de
S '12
S '22
S '26
⎡ α1 ⎤
with ⎡α xx ⎤
⎢ ⎥
-1 ⎢
⎥
⎢α yy ⎥ = [R ][T ][R ] ⎢α 2 ⎥
⎢ ⎥
⎢⎣ 0 ⎥⎦
⎣α xy ⎦
⎡β xx ⎤
S '16 ⎤ ⎡ σ1 ⎤ ⎡α xx ⎤
⎢ ⎥
⎢ ⎥
⎥
⎢
⎥
S '26 ⎥ ⎢σ 2 ⎥ + ⎢α yy ⎥ ΔT + ⎢β yy ⎥ Δc
⎢β xy ⎥
S '66 ⎥⎦ ⎢⎣σ 6 ⎥⎦ ⎢⎣α xy ⎥⎦
⎣ ⎦
and
⎡β xx ⎤
⎡ β1 ⎤
⎢ ⎥
-1 ⎢
⎥
⎢β yy ⎥ = [R ][T ][R ] ⎢β 2 ⎥
⎢ ⎥
⎢⎣ 0 ⎥⎦
⎣β xy ⎦
Mechanics of Composite Materials
Technology of Polymers and Composites &
Engineering Mechanics
Dmytro
y Vasiukov
dmytro.vasiukov@mines‐‐douai.fr
dmytro.vasiukov@mines
111
x
MATHEMATICAL PRELIMINARIES
MATHEMATICAL PRELIMINARIES
vectors
vectors
Summation:
Vector in three‐dimensional space:
A = a1e1 + a 2e 2 + a 3e3
A = a1e1 + a2e 2 + a3e3
a1, a2 , a3 components
3
A = ∑ a je j
j=1
e1, e 2 , e3 basis vectors
A = a j e j = a k e k = a me m
dummy index
Scalar product (“dot product”):
A ⋅ B = ( Aieˆ i ) ⋅ ( B j eˆ j ) = Ai B j δijj = Ai Bi
When basics vector are constant, with fixed length and
direction, the coordinate system is called Cartesian
⎧1, if i = j
δij ≡ ⎨
0 if i ≠ j
⎩ 0,
Vector product (“cross product”):
A × B = ( Aieˆ i ) × ( B j eˆ j ) = Ai B j δij = Ai Bi εijk eˆ k
When it is orthogonal, it is called rectangular Cartesian.
0, for i = j ,or j = k , or k = i
⎧
⎪
εijk ≡ ⎨ 1,
1 for
f (i, j , k ) ∈ {(1,
(1 22,3),(2,3,1),(3,1,
3) (2 3 1) (3 1 2)}
⎪ −1, for (i, j , k ) ∈ {(1,3, 2),(3, 2,1),(2,1,3)}
⎩
( x, y , z )
( x1, x2 , x3 )
113
back to slides
114
STRESS TENSOR
back to slides
STRAIN‐DISPLACEMENT
tensors
kinematics
Stress vector
ΔF
Δ
F (nˆ )
ΔS →0 ΔS
t (nˆ ) = lim
Cauchy stress tensor and Cauchy stress formula
σ = σij eˆ ieˆ j
t (nˆ ) = nˆ ⋅ σ
Displacement vector
2nd order tensor or dyad
u=x−X
σ = σ11eˆ1eˆ1 + σ12eˆ1eˆ 2 + σ13eˆ1eˆ 3
+ σ 21eˆ 2eˆ1 + σ 22eˆ 2eˆ 2 + σ 23eˆ 2eˆ 3
+ σ31eˆ 3eˆ1 + σ32eˆ 3eˆ 2 + σ33eˆ 3eˆ 3
Double dot product:
Green‐Lagrange strain tensor:
1
E = ⎡⎣∇u + (∇u)T + ∇u ⋅ (∇u)T ⎤⎦
2
A : B = ( Aij eˆ ieˆ j ) : ( Bmneˆ meˆ n ) = Aij Bmn (eˆ j ⋅ eˆ m )(eˆ i ⋅ eˆ n ) =
Infinitesimal strain tensor:
115
•
J.N. Reddy Theory and Analysis of elastic plates and Shells, 2nd ed., CRC Press, 207, p. 303
| ∇u |<< 1
if
= Aij Bmn δ jmδin = Aij B ji = Amn Bnm
back to slides
116
•
∂u
∂u ∂u
1 ⎛ ∂u
E jk = ⎜ j + k + m m
2 ⎜⎝ ∂X k ∂X j ∂X j ∂X k
1
E ≈ ε = ⎣⎡∇u + (∇u)T ⎤⎦
2
1 ⎛ ∂u
∂u
ε jk = ⎜ j + k
⎜
2 ⎝ ∂xk ∂x j
J.N. Reddy Theory and Analysis of elastic plates and Shells, 2nd ed., CRC Press
⎞
⎟⎟
⎠
⎞
⎟⎟
⎠
back to slides
GENERELIZED HOOK’S LAW AND SYMETRY OF STIFFNESS
TRANSFORMATION RULES
for vectors
C ‐ stiffness 4th order tensor ,in general case has
σ = C:ε
(3)^4=81
(3)
4 81 components. But independent ones are
components But independent ones are
considerably less (symmetry of stress and strain
tensors, existence of the strain energy).
I h b
In the absence of body couples, the principle of the conservation of angular
fb d
l
h
i i l f h
i
f
l
momentum leads to symmetry of the stress tensor:
σij = Cijkl : ε kl
σij = σ ji
( )
p
6(3)^2=54 components
Cijkl
Strain tensor symmetrical by its definition:
eˆ′i = Rij eˆ j
eˆ i = R jieˆ ′j
ε kl = εlk
eˆ ′ = Reˆ ,
eˆ = R −1eˆ ′ = R T eˆ ′
Cijkl
6x6=36 components
Transformation rule:
Strain energy density function is invariant with respect to derivatives over strain:
∂U 0
σij =
= Cijkl : ε kl
∂εij
117
•
∂U 0
= Cijkl
∂εij ∂ε kl
Cijkl
21
21 components
t
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J.N. Reddy Theory and Analysis of elastic plates and Shells, 2nd ed., CRC Press
118
R ji ≡ cos(eˆ i , eˆ ′j )
Rij ≡ cos(eˆ ′i , eˆ j )
R ‐rotation matrix
⎡ eˆ1′ ⎤ ⎡ c s ⎤ ⎡ eˆ1 ⎤
⎢eˆ ′ ⎥ = ⎢ − s c ⎥ ⎢eˆ ⎥
⎢ 2⎥ ⎢
⎥⎢ 2⎥
⎢⎣ eˆ′3 ⎥⎦ ⎢⎣
1⎥⎦ ⎢⎣ eˆ 3 ⎥⎦
TRANSFORMATION RULES
UD COMPOSITES
for tensors
use of transformation rules
use of transformation rules
Transformation rule: σ′ij = Rik R jl σ kl
2
s2
2 sc ⎤
⎡ σ1 ⎤ ⎡ c
⎢ σ ⎥ = ⎢ s 2 c 2 −2 sc ⎥
⎥
⎢ 2⎥ ⎢
⎢⎣ τ12 ⎥⎦ ⎢⎢ − sc sc c 2 − s 2 ⎥⎥
⎣
⎦
σ
Contracted notation: σ′p = T pq σq
ε
σ
−1
ε′p = T pq ε q
ε
( p, q = 1...6)
⎡ σx ⎤
⎢ ⎥
⎢σy ⎥
⎢ τ xy ⎥
⎣ ⎦
T
⎛
⎞
⎛
⎞
⎜ T pq ⎟ = ⎜ T pq ⎟
⎝
⎠
⎝
⎠
⎡ c2 s2 0
⎢ 2
c2 0
⎢s
⎢ 0
σ
0 1
T =⎢
0 0
⎢ 0
⎢ 0
0 0
⎢
⎢⎣ − sc sc 0
119
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0
0
0
c
s
0
2 sc ⎤
⎥
0
−2 sc ⎥
0
0 ⎥
⎥
−s
0 ⎥
c
0 ⎥⎥
0 c 2 − s 2 ⎥⎦
0
c = cos θ
⎡ c2
s2
⎢ 2
c2
⎢ s
⎢ 0
ε
0
T =⎢
0
⎢ 0
⎢ 0
0
⎢
⎢⎣ −2 sc 2 sc
s = sin θ
0 0
0 0
1 0
0 c
0 s
0 0
⎡ ε1 ⎤ ⎡ 1 / E1
⎢ ε ⎥ = ⎢ −ν / E
⎢ 2 ⎥ ⎢ 21 2
⎢⎣ γ12 ⎥⎦ ⎢⎣
0
sc ⎤
⎥
0
− sc ⎥
0
0 ⎥
⎥
−s
0 ⎥
c
0 ⎥⎥
0 c 2 − s 2 ⎥⎦
0
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−ν12 / E1
1 / E2
0
2
s2
− sc ⎤
⎡ εx ⎤ ⎡ c
⎥
⎢ ⎥ ⎢ 2
c2
sc ⎥
⎢ εy ⎥ = ⎢ s
2
2⎥
⎢ ⎥ ⎢
⎣ γ xy ⎦ ⎣⎢ 2 sc −2 sc c − s ⎦⎥
120
0 ⎤ ⎡ σ1 ⎤
0 ⎥ ⎢ σ2 ⎥
⎥⎢ ⎥
1 / G12 ⎥⎦ ⎢⎣ τ12 ⎥⎦
⎡ ε1 ⎤
⎢ε ⎥
⎢ 2⎥
⎢⎣ γ12 ⎥⎦
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MICROMECHANICS
Review of models:
• Voigt
• Reuss
• Hill theorem
• Bounding theorem
• The dilute approximation
h dil
i
i
• The Composite sphere assemblage
• The self‐consistent scheme
Concentric Cylinder Assemblage (CCA) Model
l d
bl
(
)
d l
•
•
•
•
•
Generalized self‐consistent scheme
Differential scheme
Mori‐Tanaka theory
Eshelby inclusion model
Numerical methods
S
Square and hexagonal packing
dh
l
ki
• Introduced by Hashin
Introduced by Hashin and Rosen
and Rosen
• Suitable for transverse isotropic materials
• 4 out of five material constants
•
•
•
121 •
Z Hashin, BW Rosen. The elastic moduli of fibre‐reinforced materials. J. Appl. Mech. 1964, Vol. 31, pp. 223‐232.
h
h l
d l ffb
f
d
l
l
h
l
RM Christensen. Mechanics of Composite Materials. Krieger Publishing Company, Florida, 1991.
http://en.wikipedia.org/wiki/Micro‐mechanics_of_failure
Jacob Aboudi, Steven M. Arnold, Brett A. Bednarcyk Micromechanics of Composite Materials:
A Generalized Multiscale Analysis Approach. 2013
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