Micromechanical modeling of strength: Long unidirectional fibers

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Micromechanical modeling of strengthLong
unidirectional fibers

• Two possible scenarios
• Matrix failure mode
• Fiber failure mode

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1. Matrix failure modeBrittle matrix with
ductile fibers Typical of ceramic matrix
composites.
2. Fiber failure modeBrittle matrix with
ductile fibers Typical of most polymer
matrix composites.
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1. Matrix failure modeBrittle matrix with
ductile fibers Typical of ceramic matrix
composites.
STRESS
sf
FIBER
COMPOSITE
sm
MATRIX
em
ef
STRAIN
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Two different failure sequences, depending on the
fiber volume fraction Vf

• Low Vf The matrix fails first, load is
  transferred to the few fibers, but fibers are
  unable to support the load so they do not reach
  their true fracture strength sf

sf
sf
sm
sf is the stress in fibers when matrix fails
0
1.0
Vf
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• High Vf The matrix fails first, load is
  transferred to the fibers, here fibers are able
  to support the load until their true fracture
  strength sf is reached

sf
sf
sm
0
1.0
Vf
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• Cross-over Vf found by combining the two
  previous equations

sf
RoM
sf
sm
0
1.0
Vf
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• EXAMPLE Glass/polyester composite

ef 0.025 and em 0.020 Therefore using sm
72 MPa, sf 2.1 GPa, Ef 76 GPa, and sf
1.52 GPa (from the stress-strain curve at ef
em), we get
OK, because most commercial laminates have Vf
0.4 to 0.7
Vf 0.11
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2. Fiber failure modeBrittle fibers in ductile
matrix Typical of polymer matrix composites.
STRESS
sf
FIBER
COMPOSITE
sm
MATRIX
ef
em
STRAIN
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Again, two different failure sequences, depending
on the fiber volume fraction Vf

• Low Vf The fibers fail first, load is
  transferred to the matrix, which is able to
  resist without failing immediately

sf
sm
0
1.0
Vf
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• High Vf The fibers fail first, load is
  transferred to the matrix, which fails
  immediately as it is unable to support the load
  (matrix true fracture strength sm is not
  reached)

sf
sm is the stress in matrix when fibers fail
sm
sm
0
1.0
Vf
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• Cross-over Vf found by combining the two
  previous equations

sf
RoM
sm
Vf
Vcrit or Vmin
0
1.0
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• EXAMPLE carbon/epoxy composite

ef 0.005 and em 0.020 Therefore using sm
80 MPa, sf 2.0 GPa, Em 5.3 GPa, and
sm 26.5 MPa, we get
Vf 0.026
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• EXAMPLE glass fiber in flexible polyester

ef 0.025 and em 0.035 Therefore using sm
65 MPa, sf 2.1 GPa, Em 2.08 GPa, and
sm 52 MPa, we get
In both examples, Vf is very small, indicating
that the matrix contribution to the longitudinal
tensile strength of polymer-based composites is
small.
Vf 0.006
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• 2. MATERIALS FOR COMPOSITES FIBERS, MATRICES
• Types and physical properties of fibers
  flexibility and compressive behavior stochastic
  variability of strength Limits of fiber
  performance types and physical properties of
  matrices combining the phases residual thermal
  stresses Lectures 4-5

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CHAPTER 2 Materials for CompositesFibers and
matrices

• Types of Fiber Reinforcement
• Glass fibers
• Carbon or Graphite Fibers
• Aramid Fibers
• Polyethylene Fibers
• Rigid Rod Polymer Fibers
• Boron Fibers
• Silicon Carbide Fibers
• Other ceramic fibers
• Prospects for future reinforcement
• Self-Reinforcing Composites Molecular
  Reinforcement
• Natural Fiber Composites
• Carbon Nanotubes

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Glass fibers

• Bulk glass tensile strength 0.7 to 1.4 GPa
• Fiber form from 3.5 to 5 GPa
• Isotropic and amorphous No crystalline order
• Used with epoxy and polyester mostly
• Susceptible to stress corrosion
• Poor fatigue resistance
• Good weather and chemical corrosion resistance
• 3D network, ionic and covalent bonds
• Sizing agent added after drawing
• Organosilane coupling agent X3SiR---matrix, X
  hydrolizes in presence of water to form silanol
  groups --- glass surface
• Density 2.54 g/cc
• Fine surface cracks leads to strength
  variability, size effects

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Amorphous structure of glass
two dimensional representation of silica glass
network
modified network (Na2O added)
Each polyhedron can be seen to be a combination
of oxygen atoms around a silicon atom bonded
together by covalent bonds. The sodium ions form
ionic bonds with charged oxygen atoms and are not
linked directly to the network. The
three-dimensional network structure of glass
results in isotropic properties of glass fibers,
in contrast to those of carbon and Kevlar aramid
fibers which are anisotropic. Thus the elastic
modulus of glass fibers measured along the fiber
axis is the same as that measured in the
transverse direction, a characteristic unique to
glass fibers.
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Composition E Glass C Glass S
Glass SiO2 55.2 65.0 65.0 Al2O3 8.0 4.0 25.0 C
aO 18.7 14.0 - MgO 4.6 3.0 10.0 Na2O 0.3 8.5 0.3
K2O 0.2 - - B2O3 7.3 5.0 -
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Sizing materials are normally coated on the
surface of glass fibers immediately after forming
as protection from mechanical damage. For glass
fibers intended for weaving, braiding or other
textile operations, the sizing usually consists
of a mixture of starch and a lubricant, which can
be removed from the fiber by burning after the
fibers have been processed into a textile
structure. For glass reinforcement used in
composites, the sizing usually contains a
coupling agent to bridge the fiber surface with
the resin matrix used in the composite. These
coupling agents are usually organosilanes with
the structure X3SiR, although sometimes titanate
and other chemical structures are used. The R
group may be able to react with a group in the
polymer of the matrix the X groups can hydrolyze
in the presence of water to form silanol groups
which can condense with the silanol groups on the
surface of the glass fibers to form siloxanes.
The organosilane coupling agents may greatly
increase the bond between the polymer matrix and
the glass fiber and are especially effective in
protecting glass fiber composites from the attack
of water.