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Parenthesis: The fragmentation phenomenon

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Under increasing load, the fiber breaks into fragments. ... THUS: fragment do not break if their length is less than this critical value, lc. ... – PowerPoint PPT presentation

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Title: Parenthesis: The fragmentation phenomenon


1
Stress profile in the fiber fragments
2
Increasing stress parallel to the fiber axis,
viewed by polarized light microscopy
3
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.

4
Focus on a fragment of a fiber
(with the option that the
fiber may be hollow)
dx
d
sfdsf
sf
D
t
5
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
6
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?
7
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
8
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).
9
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

10
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11
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

12
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.

13
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)
14
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
15
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16
In these coordinates, a straight line is obtained
if the exponential model is correct
17
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
18
Using continuously monitored fragmentation tests,
we obtain the following
saturation
saturation
19
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
20
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
21
EXPERIMENTAL DIFFICULTIES
  • Glass/epoxy in water What happens to the
    fragmentation process?

22
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23
Debonding arises also in many systems under dry
conditions
24
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.

25
Early energy balance schemes(they were never
applied to the fragmentation phenomenon
  • Mullin Mazio (1968, 1971)

26
  • 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
27
  • 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
28
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
29
Solution
There are two forms
30
Glass fiber in epoxy
31
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32
Glass fiber in epoxy
33
  • 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

34
Single-fiber fragmentation as
a
test of interfacial adhesion
Fiber break
Low strain
Single fiber embedded in matrix
higher strain
Fragmented fiber
Saturation limit
35
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?

36
  • 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
37
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
38
NORMAL LIGHT MICROSCOPY
Pre-load
No pre-load
39
POLARIZED LIGHT MICROSCOPY
Pre-load
No pre-load
40
Explanation
41
How are thermal residual stresses (which may
cause spontaneous fiber breakage) accounted
for?
The total stress on the preloaded fiber is
42
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.
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