Parenthesis: The fragmentation phenomenon

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Stress profile in the fiber fragments
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Increasing stress parallel to the fiber axis,
viewed by polarized light microscopy
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Interpretation of the test

• Under tension, the tensile stress is transferred
  to the fiber by shear stress at the fiber-matrix
  interface
• Under increasing load, the fiber breaks into
  fragments. These are successively shorter (on
  average)
• A limit is reached when the fragment length is so
  small (and therefore the fragment is so strong)
  that the applied tensile stress can no longer
  reach the strength of that fragment
• THUS fragment do not break if their length is
  less than this critical value, lc. In principle,
  all fragments should reach this length.
• Actual (measured) fragment lengths vary between
  lc and lc/2, thus a mean length of 3 lc/4.

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Focus on a fragment of a fiber
(with the option that the
fiber may be hollow)
dx
d
sfdsf
sf
D
t
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We get
Integrating
Neglecting s0 and assuming that t is constant (
ti) along the fragment length, we obtain an
expression for the interfacial shear strength
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For a full fiber (dNT 0), this becomes
This is the Kelly-Tyson expression. It provides
an interpretation of the raw data (at saturation)
of the fragmentation test (examine the effects of
s, D, and lc)
What can be said about the fragment length
distribution at the saturation limit?
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Kelly-Tyson shows that the fiber strength is a
linear function of the fragment length (and its
slope is a constant, t). Thus, there exists a
certain length (the critical length) such that
the applied stress becomes just equal to the
fragment strength
s
s
Fiber strength
Fiber strength
Fiber stress
Fiber stress
Fragment length llc One more break leads to 2
fragments of length lc/2
Fragment length lfragment of length lc
The (measured) fragment lengths vary between lc
and lc/2, thus an average fragment length of
3lc/4. Actually, the window of fragment
lengths is even wider than 21, it is more like
41, due to the fiber strength distribution
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If we insert this into the K-T equation we obtain
quantitative interpretation of the
fragmentation test
Example Graphite/epoxy single-fiber
composite Measurements show that the strength
and average fragment length at saturation, and
fiber diameter, are respectively 4 GPa, 150
micron and 10 micron. Matrix is epoxy. Thus, the
interface strength is equal to 100 MPa, a very
large value. But how is fragment strength at
saturation measured? The same fiber in
polypropylene reveals fragments of average length
1000 micron ( 1 mm), which reflects a much
poorer adhesion (3.8 MPa).
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How is a fragmentation (or SFC) test performed
and how is sf(lc) measured?

• A blind test is performed in a tensile tester
  until the fragmentation saturation limit is
  reached. The fiber diameter and average fragment
  length at saturation are measured by optical
  microscopy under polarized light.
• The strength of very short fragments, at the
  saturation limit, is obtained by extrapolating
  from the strength of fibers tested at longer
  lengths

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Early 90sNew experimental schemes using the
fragmentation test

• CONTINUOUS MONITORING of the fragmentation test
  By following the progression of the fragmentation
  phenomenon under the microscope, new and easily
  collected data is obtained
• The experimental fragment length distribution
• The Weibull parameters of the single fiber and
  the scale effect are obtained from a single test
• The interfacial strength, using data from first
  breaks rather than at saturation
• The interfacial energy, using a new model

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CONTINUOUS MONITORING
1. Distribution of fragment lengths

• Traditionally the distribution of the fragment
  lengths at saturation is assumed to be either
  Weibull, or normal (Gaussian), or lognormal,
  based on good fit.
• In fact, it is easily shown that the exact
  fragment length distribution below saturation is
  an exponential distribution
• It is still approximately so at saturation.
• See Wagner Eitan, Applied Physics Letters 56
  (1990), p. 1965
• Demonstration Assume that the defects are spread
  on the fibers according to a spatial Poisson
  process there are no overlaps of defects, and
  there is independence of position.

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The probability of finding k defects along a
length s is
(l 0)
The probability of finding 0 defects along a
length s is therefore
The cdf for s is therefore
But at the defect site there is a small zone of
length lc/2 (on each side) where a defect, if
present, will almost surely not be activated
(why?). Thus, to account for this forbidden
zone, we have
(1/l average fragment length at a given stress,
and m is proportional to lc)
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Thus, away from saturation, the fragment length
distribution is exponential (called a shifted
exponential). This is correct as long as the
critical zone is much smaller than the average
spacing between breaks because the defect
spreading process is truly random (truly
Poisson). This is not so at saturation, the data
deviate from the exponential form. Example
Carbon/epoxy fragmentation
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In these coordinates, a straight line is obtained
if the exponential model is correct
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2. Effect of fragment length on strength A second
interesting result concerns the size effect. We
remember that, traditionally, extensive
experimentation is necessary to determine the
strength of very short fragments by
extrapolation
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Using continuously monitored fragmentation tests,
we obtain the following
saturation
saturation
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We know that the mean tensile strength of fibers
of length L is
Where a and b are the Weibull scale and shape
parameters for strength. Assuming that the
fragmentation test may be viewed as a multiple
tensile test with independent samples of varying
lengths, for which Weibull statistics applies, we
apply the equation above (in reverse form) to
obtain a relationship between the average
fragment length () as a function of fiber
stress
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This describes well the experimental plots
presented earlier (and note that sf (Ef/Em)sm).
This relationship is valid away from saturation.
At saturation, the fragment lengths becomes
insensitive to the applied stress. Why is this
important? Because this method provides a simple
way to measure the strength of very short
fragment lengths, instead of extrapolations. More
over, from the linear slope we obtain the shape
parameter of the fiber, b, The scale parameter a
can be calculated from the intercept which is
equal to
Reference Yavin et al., Polymer Composites, 12
(6) December 1991, 436
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EXPERIMENTAL DIFFICULTIES

• Glass/epoxy in water What happens to the
  fragmentation process?

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Debonding arises also in many systems under dry
conditions
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Why interfacial debonding is a problem

• Because it has a biasing effect on the saturation
  limit (there are less, longer, fragments than
  there should be)
• Because the Kelly-Tyson model does not account
  for debonding and thus, is not valid when
  debonding is present
• Because it is extensive under humid conditions
• In the early 90s, the stress-based approach and
  analysis of Kelly-Tyson was therefore found to be
  difficult to apply when debonding is present. A
  new approach was developed, based on an energy
  balance model.

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Early energy balance schemes(they were never
applied to the fragmentation phenomenon

• Mullin Mazio (1968, 1971)

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• Outwater Murphy (1969)
• Same geometry as a compressive fragmentation
  test. The formula works in tension as well

Where x is the debonding length counting one side
of the break. For very short interfacial cracks,
x 0 and the limiting energy value is
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• DiBenedettos model for fragmentation (1991)
• New model developed at WIS (1993-1995)

PRINCIPLE When a fiber breaks, the strain energy
in the length Ldd prior to the break, is
expended in the formation of fiber fracture
surfaces, as well as in the formation of debonded
surfaces at the interface
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Energy balance1. H.D. Wagner, J.A. Nairn, M.
Detassis, Applied Composite Materials, 2 (2)
(1995), 107-117.2. M. Detassis, E. Frydman, D.
Vrieling, X.-F. Zhou, J. A. Nairn, H. D. Wagner,
Composites, Part A 27A (1996), 769-773.

• The energy available is thus
• The energy that must be expended to form fresh
  fiber and interface fracture surfaces is

The following energy balance equation is obtained
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Solution
There are two forms
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Glass fiber in epoxy
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Glass fiber in epoxy
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• Advantages of energy balance model
• Debonding used positively
• Information from first breaks
• No need to reach saturation
• No need for strength of short fragments
• More universal (?) than stress-based approach

• Disadvantages of energy balance model
• Debonding is necessary
• Shear-lag model, need for b
• Linear elastic model
• Some energy dissipative processes cannot be
  accounted for

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Single-fiber fragmentation as
a
test of interfacial adhesion
Fiber break
Low strain
Single fiber embedded in matrix
higher strain
Fragmented fiber
Saturation limit
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Two issues of importance

• In some important composite materials, such as
  glass-epoxy, the saturation limit (the critical
  fragment length) is difficult to reach because
  the strain to failure of the fiber is very close
  to that of the matrix e(glass) e(epoxy)
  0.04. Can the fragmentation test still be useful
  for such composites? How can we force the
  fragmentation phenomenon to occur below the
  saturation limit?
• 2. In other important composites (HM
  graphite/epoxy), thermal residual stresses are
  very high and following specimen preparation, the
  fiber is broken into fragment prior to testing!
  How are these stresses being accounted for?

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• e(glass) e(epoxy) 0.04.
• Can we force fragmentation, and thus get info
  about interfacial adhesion?
• Can the fragmentation test still be used?

Yes, provided the fiber is pre-loaded with
calibrated weights. For demonstration of the
technique, we use a twin-fiber specimen
spre
spreload
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The two identical glass fibers are separated by a
relatively large distance, and embedded in
polymer film The film is polymerized, then
separated from its substrate, the weights are
removed The twin-fiber specimen is loaded in
tension exactly like in a monitored fragmentation
test, under polarized light microscopy The
stress-strain curve is obtained together with the
observed fiber breakage sequence
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NORMAL LIGHT MICROSCOPY
Pre-load
No pre-load
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POLARIZED LIGHT MICROSCOPY
Pre-load
No pre-load
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Explanation
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How are thermal residual stresses (which may
cause spontaneous fiber breakage) accounted
for?
The total stress on the preloaded fiber is
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Conclusions

• If a stress-based (K-T) interpretation is used
  for the test, the fiber pre-loading technique
  provides a way to reach saturation.
• If an energy-based interpretation is used, the
  fiber pre-loading technique provides a way to
  work within the linear portion of the
  stress-strain curve.
• In both cases, it is critically important to know
  the exact stress level in the fiber to get a
  correct interpretation of the test.