Resistance Heating for Self-healing
Composites
NATHAN KWOK* AND H. THOMAS HAHN
Department of Mechanical and Aerospace Engineering
University of California, Los Angeles
Los Angeles, CA 90095, USA
ABSTRACT: Thermally remendable polymers have recently become available that
are paving the way for the creation of self-healing composites. This article
investigates the use of carbon fibers as a resistive heating network for future use
in a self-healing composite. Various electrode methods are studied in an attempt
to reduce the contact resistance that causes localized heating around the electrode.
In addition to localized heating, electrical conduction through the fibers creates
heat through composite resistance. It was observed that for short distances between
electrodes, contact resistance dominates while over longer distances, composite
resistance dominates. The balance between the two types of heating is measured
and modeled using finite element modeling (FEM).
KEY WORDS: self-healing composite, resistance heating, electrical resistance,
finite element modeling.
INTRODUCTION
R
ESISTIVE HEATING HAS many applications and is most commonly seen in electric
stoves and heaters. It operates on the principle that current passing through a resistive
element must dissipate power according to the equation P ¼ I2R, where P ¼ power (W),
I ¼ current (A), and R ¼ resistance (). Heating can be provided in composites by treating
the fibers, which are conducting, as resistive elements while the surrounding matrix acts as
an insulator. This is applicable in the use of a specially made, heat-activated resin that
utilizes a Diels–Alder reaction to heal itself [1]. When this resin is combined with carbon
fibers, a self-healing composite can result. The challenge lies in efficiently delivering the
heat, without needlessly heating areas far from the crack.
There are two kinds of resistance: composite and contact. Composite resistance is
associated with the resistivity of the individual fibers, a material property, as well as the
*Author to whom correspondence should be addressed. E-mail: natek@ucla.edu
Figures 1–3, 5–16 and 18–22 appear in color online: http://jcm.sagepub.com
Journal of COMPOSITE MATERIALS, Vol. 41, No. 13/2007
0021-9983/07/13 1635–20 $10.00/0
DOI: 10.1177/0021998306069876
ß 2007 SAGE Publications
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1635
1636
N. KWOK AND H. T. HAHN
length and diameter of the fibers. In addition, electrons pass internally between adjacent
fibers in contact. This internal contact resistance is associated with the contact surface area
between two fibers in contact. Because the matrix is almost insulating, most conduction
occurs through fibers. These resistances combine to create a highly anisotropic composite
resistance. The internal fiber structure can be modeled as a network of interconnected
resistors [2], with the resistors running along the fiber direction representing fiber
resistance, and resistors running against the fiber direction representing internal contact
and matrix resistance.
In addition, there is significant external contact resistance between the electrodes for
the external power source and the outer surface of the composite itself. This is a function
of the method of electrode attachment as well as the presence of the resin-rich upper layer
that can interfere with fiber/electrode contact. Although very low contact resistances
can be achieved through copper electroplating [3], an attempt is made here to achieve
low contact resistance using simpler and more portable methods. Seven different electrode
configurations are tested to determine their contact resistance. This external contact
resistance is strictly between the electrode and the fiber surface, and is not to be confused
with the internal contact resistance between individual fibers, which is a component
of composite resistance and is not analyzed separately. The resistance is measured directly
using a sourcemeter. Resistive heating experiments are also performed using a DC power
supply and IR thermometer.
A finite element analysis is done using the FEMLab to simulate the effects of resistive
heating. The parameters of the simulation are made identical to those of the actual
specimens to test the validity of the model.
EXPERIMENTAL PROCEDURE FOR RESISTANCE MEASUREMENT
Two laminates were used in testing, both prepared from 255 255 mm sheets of
AS4/3502 prepreg, one with a [0/90]S layup (four layers, ‘Composite 1’) and another with
a [90/0]S layup (‘Composite 2’). They were autoclave cured at 450 K (350 F) under
0.55 MPa (80 psi). The cured laminates were then cut into strips, with each strip measuring
255 21 0.68 mm. Some of the tests required the specimens to be sanded to remove the
resin rich top layer. This was done with 180 grit sand paper until it began to turn black,
indicating the fibers were exposed, and then finish sanding was performed using 400 grit
followed by 1000 grit. Once sanding was done, the thickness was reduced to 0.62 mm
for Composite 1 and 0.61 mm for Composite 2.
To test the contact resistance, a Keithley sourcemeter was used in two-probe mode at
2.1 V (Figure 1). The meter measures resistance by applying a DC voltage and measuring
the current that passes through the specimen. The meter was connected to the sample using
alligator clips, either alone or using one of the six electrodes as an intermediary.
The materials used in the intermediate electrodes are listed in Table 1, and the electrode
configurations are listed in Table 2.
Electrodes 1, 2, and 7 can be considered ‘moving’ electrodes in that they are in no way
permanently attached to the composite. This allows for the distance between electrodes
to be varied in small intervals (5 mm intervals were used), resulting in many test points.
The design for electrodes 2 and 7 is illustrated in Figure 2. These are referred to as
‘contact electrodes’ because they utilize a C-clamp to apply contact pressure on the surface.
The contact electrodes were constructed in an attempt to lower contact resistance while still
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1637
Resistance Heating for Self-healing Composites
Resistance measurement composite 1: [0/90]s
Sourcemeter
Top surface sanded
0.62 mm
d
21 mm
255 mm
Vise grip
Rubber
insulation
Bottom surface
insulated by
electrical tape
Alligator
clip
Figure 1. Sample held in place by clamp and measured using a sourcemeter or LCR. Alligator clips attach
either directly to bare composite or to an intermediate electrode.
Table 1. List of electrode materials.
Material name
Manufacturer
Catalog no.
Copper tape
Carbon tape
Silver paint
Copper sheet
Conductive rubber
Miniature C-clamp
SPI Supplies
SPI Supplies
SPI Supplies
McMaster Carr
Zoflex
Nelson Hobby
05012-AB
05081-AB
05001-AB
8944K37
CD45.1-6S-2
C100 and C102
Table 2. List of electrode types.
No.
Material
Contact size
Measurement distance
intervals, d (mm)
1
2
None (alligator clip on bare composite)
Small conducting rubber pad, silver paint,
copper sheet, C-clamp
Single-sided adhesive copper tape
Double-sided adhesive carbon tape
Narrow silver paint line
Wide silver paint line
Large conducting rubber pad, silver paint,
copper sheet, C-clamp
Multiple point contacts
6-mm diameter (circular)
5–235 at 5 mm intervals
10–230 at 5 mm intervals
6 6.5 mm
6 21 mm
6 21 mm
12 21 mm
10 12 mm
5, 100, 200
5, 100, 200
5, 10, 20, 50, 100
20, 50, 100
10–220 at 5 mm intervals
3
4
5
6
7
allowing for portability. The contact surface is composed of a conducting rubber pad made
by Zoflex, with a small round version measuring 6 mm in diameter and 0.5 mm in thickness
and a large rectangular version measuring 10 12 mm. This pad was then wetted with silver
paint and attached to a thin copper plate measuring 6 20 mm for the small version and
10 20 mm for the large version. The plate was then glued to a lightweight plastic C-clamp.
The entire assembly allows for multiple electrodes to be attached to the copper plate via
alligator clips and then easily loosened and moved from location to location.
Electrodes 3–6 were fixed on the composite. This limited the number of distance
intervals because too many electrodes could effectively shorten the length of the
composites by creating low resistance zones. For the copper and carbon tape (3 and 4),
the electrodes were cut to size and pressed onto the surface of the composite. In both cases
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1638
N. KWOK AND H. T. HAHN
Alligator clip
Contact electrode
Composite
Rubber conducting pad
Silver paint interface
Copper sheet
Alligator clip
Composite
C-Clamp
Electrical tape
Crazy glue
Figure 2. Alligator clips are a quick method of attachment but have a small contact area. Electrode 2,
‘small contact electrode’ can act as an intermediary and is a better design because it applies pressure
to the surface and has more contact area.
5
10
mm
mm
20 mm
50 mm
100 mm
Figure 3. The thickness of the fixed electrodes limited the interval spacing.
the adhesive was designed for such an application and is electrically conductive. In the case
of the silver paint strips (5 and 6), these were painted on with a small brush and allowed
to dry. Figure 3 is a photograph of electrode type 5: narrow silver paint lines.
EXPERIMENTAL PROCEDURE FOR RESISTANCE HEATING
The circuit setup and temperature measurement apparatus for resistive heating
are shown in Figure 4. The purpose is to use electricity to heat the composite and
measure the resulting temperature field. Two methods of electrical contact were used,
Figure 4: the simple alligator clip, (electrode 1 from Table 2) and the ‘large contact
electrode’ (electrode 7 from Table 2).
To measure temperature, a Land Instruments PockeTherm 32C IR thermometer was
used with the measurement point occurring every 10 mm, indicated by the dots in Figure 4.
The thermometer was calibrated using a thermocouple. To minimize conduction out of the
specimen, the specimen was clamped in place by a small vice lined with ceramic insulation.
RESULTS: CONTACT RESISTANCE
The overall resistance R measured by the Sourcemeter can be modeled as a series
of three resistors with the equation:
R ¼ R1 þ R2 þ R3
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
ð1Þ
1639
Resistance Heating for Self-healing Composites
Resistively heating composite 1: [0/90]S
DC power
supply
Ammeter
x
I+
I−
170 mm
Infrared
thermometer
21 mm
20 mm
V+
V−
Voltmeter
Ceramic
insulation
255 mm
Figure 4. Dots represent thermal measurement points. lþ/l and Vþ/V are the attachment points for the
power supply and voltmeter, respectively.
where R1 and R3 are the contact resistances associated with the two electrodes. We can
assume that R1 ¼ R3 because in every case, the positive and negative electrodes are
identical. R2 is the resistance of the composite given by
R2 ¼
d
A
ð2Þ
where is the effective resistivity of the composite, d is the distance between two
electrodes, and A is the cross-sectional area. This equation depends on the assumption that
potential remains constant on a cross section and only varies in the axial direction, and
is reasonable for a long, thin specimen like the one used. The resistance values should
scale linearly as the electrode spacing increases, since the resistivity and cross-sectional
area remain constant. While composite resistance scales linearly, the contact resistances
R1 and R3 should remain constant as long as the electrode type remains the same. Thus we
can solve Equation (1) and (2) for R1 by plotting R(d) at multiple values of d, and using
linear regression to arrive at a y-intercept, which is the value of 2R1.
Alligator Clips
The first test was conducted on Composite 1 with no intermediate electrode, only
alligator clips clamped onto a bare composite. The distance was varied from 5 mm
to 245 mm at 5 mm intervals. At each interval, the resistance was recorded and is shown
in Figure 5.
Although the Sourcemeter can read resistance accurately down to 1 m, the precision
was limited to 0.1 due to fluctuations caused by the electrode itself. For this electrode
there was also extremely poor consistency. Although the reading did appear to trend
upward, the residual standard deviation (also known as the standard error of the
y estimate) of the measurements was 3.86 , too large to make a conclusion. In addition to
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1640
N. KWOK AND H. T. HAHN
40.0
35.0
Resistance (ohms)
30.0
25.0
y = 0.0213x + 24.935
20.0
15.0
10.0
5.0
0.0
0
50
100
150
Distance (mm)
200
250
Figure 5. Resistance varied wildly without an intermediate material.
poor consistency, the y-intercept of the linear regression was 24.9 , indicating a contact
resistance (R1 þ R3) so large that it dwarfed any composite resistance (R2). The problem
lies in the irregularity of the composite surface and the low conductivity of the composite.
In addition, even careful sanding can still leave some resin on the surface of the composite
which can interfere with the very small contact areas of an alligator clip. By contrast,
attaching an alligator clip to a highly conductive copper plate always ensures a good
contact. Clearly, an intermediate material would be beneficial.
Small Contact Electrode
Utilizing the contact electrode shown in Figure 2 greatly improved the results, both
in terms of measurement consistency and contact resistance. This time the residual
standard deviation was only 0.10 , a 97% reduction. The equation for residual standard
deviation is:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
ðyi y^i Þ2
sy ¼
n2
ð3Þ
Where yi is the measured y-value, ŷi is the fitted y-value, and n is the number of samples.
The resulting equation for resistance is
Rðd Þ ¼ 0:0053d þ 1:3178
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
ð4Þ
Resistance Heating for Self-healing Composites
1641
so that the contact resistance of each electrode is:
R1 ¼ R3 ¼
1:3178
¼ 0:659 :
2
ð5Þ
The composite resistivity, , can also be calculated from Equations (2)–(4) in conjunction
with the measured cross-sectional area 13.02 mm2:
¼ 0:0069 -cm
ð6Þ
This result compares favorably with the measured value from the literature [3] of
0.00293 -cm for unidirectional composite under the assumption that no current passes
through the 90-degree plies. In reality, a small fraction of current does pass through the
90-degree plies, but the ratio of conductivity between the 0 and 90 directions has been
measured to be as high as 1400:1 [3] and as low as 26:1 [4], meaning even in the best
case the amount is very small.
The contact resistance for the contact electrode was lower due to a combination of
a larger contact area and the rubber pad’s ability to conform to the micron size asperities
of the surface. In addition, the C-clamp exerted a downward pressure on the electrode,
further improving its performance. By contrast, although the alligator clips also exerted
a downward pressure, it had a small net contact area and thus a lower likelihood of
contacting any exposed fibers. Interestingly, the contact electrode still utilized alligator
clips; however, in this case the alligator clips were attached to a copper sheet instead of
directly to the composite. Considering that there were more contact interfaces involved
in the use of the contact electrode, one might think this would actually increase contact
resistance; however, this was not the case. In fact, the additional contact interfaces actually
had a negligible effect on overall contact resistance. Clearly, the critical interface is
between the fiber surface and whatever it is in contact with.
The composite resistance determined from Figure 6 can be used as a baseline for other
electrode materials that do not allow for as many measurements. Since the composite
resistance does not change when different electrodes are used, it can simply be subtracted
from the total resistance to get contact resistances of other electrodes.
Copper, Carbon, and Silver Electrodes
The ‘fixed’ type electrodes were either conductive tapes (copper and carbon) or
conductive paint (silver). As shown in Figure 3, this allowed for limited measurement
intervals. The results are summarized in Figure 7. Results from the small contact electrode
as well as the big contact electrode, which utilized a larger rubber pad, are also included.
The carbon tape, which is made by SPI Supplies and intended for attaching specimens
to scanning electron microscope holders, is designed to provide a conducting
path from the specimen (which must be conducting) to the specimen holder (which acts
as a ground). For the present purpose, it performed very poorly, actually increasing
the contact resistance compared to the alligator contact by itself, in addition to being
wildly erratic.
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1642
N. KWOK AND H. T. HAHN
3.0
Resistance (ohms)
2.5
R(d ) = 0.0053d + 1.32
2.0
1.5
1.0
0.5
0.0
0
50
100
150
Distance (mm)
200
250
Figure 6. Measurement results greatly improved.
50
Small contact
Alligator
Copper
Carbon
Narrow silver
Wide silver
Big contact
Resistance (ohms)
40
30
20
10
0
0
50
100
150
Distance (mm)
200
250
Figure 7. Complete summary of resistance.
The adhesive-backed copper tape, which was also designed for the same purpose as the
carbon tape, did much better, markedly reducing contact resistance as well as showing
reasonable consistency, although it still fell well short of the contact electrode.
The silver paint was better than the copper tape but also still short of the contact
electrode. Surprisingly, when two versions of the silver paint were tried, one using
a narrow 6 21 mm painted area and one using a double size 12 21 mm area, the narrow
electrode performed better, nearly matching the contact electrode. One possible
explanation is that the silver was applied in a thinner layer for the wider electrode.
The reverse result occurred when increasing the size of the contact electrode: the total
contact resistance (R1 þ R3) was reduced from 1.318 to 0.788 while still maintaining
the same excellent consistency. As expected, the measured composite resistance (R2) was
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1643
Resistance Heating for Self-healing Composites
Contact resistance of various electrodes for composite 1
35
32.59
30
Resistance (ohms)
25
Double-sided carbon tape
Bare composite
24.78
Single-sided copper tape
Wide silver paint line
20
Narrow silver paint line
Small contact electrode
15
Large contact electrode
9.13
10
5.67
5
2.52
1.32
0.788
0
Method
Figure 8. Contact resistance values varied greatly.
almost identical. To calculate the contact resistance values listed in Figure 8, the distance,
d, at each measurement interval was multiplied by 0.053 /mm, Equation (3) to arrive
at the appropriate composite resistance. This was then subtracted from the total resistance
measured to obtain the contact resistance. The resulting contact resistance values at each
interval were then averaged.
Comparison Between [0/90]S and [90/0]S Laminates
In addition to comparing electrodes, a second composite laminate, Composite 2, was
made identically to Composite 1 with the only difference being that it was cut such that
the top and bottom layer of fibers ran in the transverse rather than longitudinal direction.
The only other difference was a very slight difference in dimensions. The purpose of this
experiment was to verify that the effects of current spreading in the thickness direction
are negligible enough that the layup does not affect the overall resistivity, given that the
number of transverse and longitudinal layers remains the same. As the thickness goes up,
this could cease to be the case. This time the test was done using the large contact electrode
since the previous test demonstrated it had the lowest contact resistance.
The results are as expected because in the short intervals, Composite 2 displays slightly
higher resistances along with less consistent results, until around 100 mm when it begins
matching Composite 1 (Figure 9). From that point on, the results are almost identical.
One explanation for the discrepancy at the shorter intervals is that the electrodes
are connected to a 90 layer which has lower conductivity. As the current spreads out,
the effect of electrode location diminishes.
Effects of Sanding Top Surface
Thus far every test has been performed on a fully sanded specimen. It is known that
by sanding the resin rich top layer, more fibers are exposed, improving conductivity.
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1644
N. KWOK AND H. T. HAHN
2.5
Resistance (ohms)
2.0
1.5
1.0
[0/90]
[90/0]
0.5
0.0
0
50
100
150
Distance (mm)
200
250
Figure 9. The [90/0] specimen displayed some inconsistency in the shorter intervals. The large contact
electrode was used in both specimens.
Figure 10. Microscopy reveals a resin rich top layer that must be removed by sanding. Magnification is 100,
FOV is 2.14 mm wide. Left: before, right: after.
To quantify this, another strip of the same dimensions was cut from Composite 1 and
resistance tested at various levels of sanding. The ‘large contact electrode’ was used in
every case, and three replicates were performed at each location to assess testing variance.
The test was done before sanding, then after sanding with 180, 320, 400, and 1000 grit for
5 min each. Photographs at 100 comparing a sanded and an unsanded sample are
shown in Figure 10. The resin-rich layer with an imprint left from the peel-ply is clearly
seen before sanding, and disappears after sanding with 180 grit. Additional sanding was
not evident in the photograph although a continual reduction in contact resistance occurs
as more sanding is performed. The other major benefit of sanding is that it greatly
improves the consistency of the results. Figure 11 shows the plot of resistances and the
lines of best fit while Figure 12 shows the plot of standard deviations. Standard deviation
decreases greatly after the first sanding, and then increases slightly before dropping again.
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1645
Resistance Heating for Self-healing Composites
3.25
Unsanded
180 grit
320 grit
400 grit
1000 grit
Resistance (ohms)
2.75
2.25
1.75
R(d )0 = 0.003812d + 1.5942
R(d )180 = 0.004431d + 1.1301
1.25
R(d )320 = 0.004585x + 1.1097
R(d )400 = 0.004658x + 0.9676
R(d )1000 = 0.004747x + 0.8846
0.75
0
50
100
Distance (mm)
150
200
Figure 11. Contact resistance (y-intercept) steadily declines as a result of sanding.
0.3
Contact resistance residual standard deviation
Ohms
0.2
0.1
0
0
180
320
Sandpaper grit
400
1000
Figure 12. Residual standard deviation makes a large improvement after first sanding.
Atomic Force Microscope
After sanding was finished, the surface was analyzed with a Digital Instruments
Nanoscope atomic force microscope at three random locations, Figure 13. The scan area
was 50 50 mm, the scan rate was 0.3 Hz, and the number of samples was 256. As the
micrographs show, sanding does a good job of exposing the fibers, which appear as
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1646
N. KWOK AND H. T. HAHN
10
20
30
10
µm
40
x 10.000 µm/div
z 4000.000 nm/div
20
30
40
10
µm
x 10.000 µm/div
z 4000.001 nm/div
20
30
40
µm
x 10.000 µm/div
z 4000.001 nm/div
Figure 13. Image scale is 10 m per division in the x/y directions, and 4 m per division in the z direction.
200
Temperature (C)
150
Alligator at 0.8 A
Contact at 0.8 A
Contact at 4.0 A
100
50
Electrode
0
0
50
Electrode
100
150
Position (mm)
200
250
Figure 14. Using alligator clips limited total current due to localized heating, while the contact electrode
performed much better.
smooth cylinders sticking slightly above the surface. This occurs because the fibers are
much harder than the matrix, causing the matrix to wear away while the fibers remain
intact. Despite this, there are still some areas where the matrix could potentially interfere
by sticking slightly above the nearest fiber. This in part explains why a more pliable
electrode such as the rubber contacts, or a liquid such as the silver paint, would perform
better than a tape or copper sheet.
MEASUREMENT OF TEMPERATURE FIELD
A DC power supply was used together with an ammeter in series and a voltmeter
in parallel to monitor current and voltage (Figure 4). Current was controlled while voltage
was left to vary depending on overall resistance. The test was done on the sanded surface
of Composite 1. Power was applied for 5 min to allow the sample to reach steady state,
and measurements were then made. Another 5 min were allowed to pass before a second
measurement, used to verify that the sample had indeed reached steady-state.
Figure 14 compares the usage of the alligator clips to the large contact electrode.
When using the alligator clips, 0.8 A was the practical current limit because anything
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1647
Resistance Heating for Self-healing Composites
2.20
Resistance (ohms)
2.15
Resistance vs temperature
at centerpoint
2.10
2.05
2.00
1.95
1.90
20
30
40
50
Temperature (C)
60
70
Figure 15. Resistance vs temperature, measured at midpoint.
higher resulted in burning the composite under the electrode. This is evident by the
extreme temperature spike in the figure, which actually underreports the true localized
temperature because the measurement interval skips over the area directly under the
electrode. By contrast, the contact electrode eliminated any noticeable spike while still
heating the area between the electrodes. This allowed the current to be raised all the way
to 4.0 A without causing burns. Some localized heating still occurs due to contact
resistance, but the elevated area between the electrodes is maintained at nearly the same
temperature. These temperatures are enough to activate the healing reaction of the
thermally mendable polymer. Outside of the test region, the temperature quickly drops
to room temperature (24 C) because there is little current flow.
RESISTANCE AS A FUNCTION OF TEMPERATURE
Figure 15 plots resistance as a function of temperature with the temperature
measurement occurring at the centerline and the resistance measured across the full
length of the composite. Although the results are somewhat inconsistent, a clear trend of
decreasing resistance as temperature increases is noted, indicating the semi-conductor-like
behavior of the composite. A total decrease of 9.2% occurred between 23 and 50 C, after
which the change appeared to level off.
MODELING WITH FEMLAB
The temperature field resulting from resistance heating under 4 A current input is
predicted using the FEMLAB [5]. All the physical parameters are made as identical to the
actual experiment as possible. The contact electrode is modeled as a very thin layer of
material with a very high electrical resistivity. This will create a large voltage drop across
a small distance and mimic the effect of contact resistance. The composite itself is modeled
in four layers using the appropriate thermal and electrical conductivities.
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
N. KWOK AND H. T. HAHN
1648
Governing Equations
The governing field equations for the steady-state heat and electrical conduction are:
r ðkrT Þ ¼ Q
r ðrrV Þ ¼ 0
ð7Þ
ð8Þ
where k is the thermal conductivity tensor, T is the temperature, V is the voltage, r is the
electrical conductivity tensor, and Q is the resistive heat source. The resistive heat source
is given by Joule’s law:
Q¼EJ
ð9Þ
J ¼ E ¼ rV
ð10Þ
where E is the electric field and J is the current density.
The appropriate boundary conditions are taken as:
n ðkrTÞ ¼ hðT Tamb Þ þ e B ðT4 T4amb Þ : convection and radiation on surfaces ð11Þ
n J ¼ 4 A : current through positive and negative electrode areas
V ¼ 0 V : negative electrode area is grounded
ð12Þ
ð13Þ
n J ¼ 0 : all other surfaces electrically insulated
ð14Þ
Here h is the convective heat transfer coefficient, Tamb is the ambient temperature, e is the
emissivity, and B is Boltzmann’s constant. For simplicity the clamps are approximated
by 10 mm tall stainless steel cubes with the same footprint as the rubber electrodes
(Figure 16).
The electrode itself has its own set of material properties and is treated as a separate
subdomain from the clamp. In addition, there is also a very thin ‘resistive layer’ that
simulates the effects of contact resistance.
The electromagnetic subdomains are analogous to the heat transfer subdomains.
Each subdomain is treated as homogeneous and there is no internal current generation.
The electrical boundary conditions are different from the thermal ones, with all
the composite boundaries acting as electrical insulators except where the two electrodes
are located.
MATERIAL PROPERTIES
Thermal
The constants used in the analysis are summarized in Table 3. The conductive heat
transfer coefficient for the composite in the 0-degree (k0) and 90-degree (k90) directions are
taken from James [6], and for the rubber electrode, k2, and stainless steel clamp, k3, from
Mills [7]. The type of rubber used was neoprene although the actual composition of the
Zoflex conducting rubber is proprietary. The emissivity, e, was measured using the
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1649
Resistance Heating for Self-healing Composites
Max:6.0
6 6
Boundary: Electric potential
5
Clamp
4
V
3
2
Electrode
Resistive
layer
1
0
0
Figure 16. Positive electrode top surface boundary condition is I ¼ 4.0 A while negative electrode top
surface is V ¼ 0 V, meaning a total of 4 A flows into the positive electrode while the negative electrode serves
as a ground. In addition to having special electrical boundary conditions (all other boundaries are electrically
insulated), the electrode top surfaces are also thermally conductive rather than convective, because in reality
the rubber electrode is attached to a C-clamp (Figure 2).
Table 3. Table of constants.
Sym
Value
Units
Description
k0
k90
k2
k3
h
e
kA
g
T
120
2.6
0.19
15.0
11.42
0.9
0.0267
0.0025
15.66 106
9.81
96
W/(m*K)
W/(m*K)
W/(m*K)
W/(m*K)
W/(m2*K)
L
B
Tinf
Tamb
C
E
0
90
t
0.021
5.67 108
297
297
2.115
1000
18282
11.53
1.61
m
J/(s*m2*K4)
K
K
(-m)1
(-m)1
(-m)1
(-m)1
(-m)1
0-deg thermal conductivity of composite
90-deg thermal conductivity of composite
Thermal conductivity of neoprene rubber
Thermal conductivity of stainless steel
Convective heat transfer coefficient
Emissivity
Thermal conductivity of air at 300 K
Thermal coefficient of volume expansion
Kinematic viscosity of air
Acceleration of gravity
Average temperature difference between
air and surface used for convective calculation
Width of slab
Boltzmann’s constant
Ambient temperature for convection
Ambient temperature for radiation
Calculated effective contact conductivity
Electrode conductivity from Zoflex [8]
Simulation conductivity 0-deg
Simulation conductivity 90-deg
Simulation conductivity thickness
W/(m*K)
K1
m2/s
m/s2
K
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1650
N. KWOK AND H. T. HAHN
infrared thermometer in conjunction with a thermocouple. The infrared thermometer
can utilize an adjustable emissivity reading, and this value was adjusted until the readout
on the IR thermometer matched the thermocouple readout. The result is consistent with
typical dark colored surfaces including other examples of the same material. The ambient
temperature for convection and radiation was simply the measured room temperature.
The convective heat transfer coefficient h is calculated by treating the sample as a heated
horizontal plate facing up undergoing natural convection. It is given by:
kA
h¼
NuL
L
NuL ¼ 0:52Ra1=4
L ¼ 0:52 ð15Þ
TgL3
2
1=4
ð16Þ
This results in a heat transfer coefficient of 11.42 W/m2 K, which is typical for natural
convection [7]. Note that the value T has to be found using iteration, using the
experimental data as a starting point.
Electrical
Unidirectional laminate strips were made using the same methods as their bidirectional
counterparts and measured for both contact resistance and composite resistivity in all
three directions. These measured values were then applied to the individual layers within
the simulation, which was for a bidirectional layup.
The effective conductivity of the contact interface is found by using the measured
contact resistance, and treating the interface as consisting of a very thin layer 0.1 mm thick.
The result is:
¼
L
0:0001
1
¼
¼ 2:115
RA 0:394 0:012 0:01
-m
ð17Þ
MODEL RESULTS
A 3-D mesh was used as shown in Figure 17. The model geometry statistics and mesh
statistics are shown in Tables 4 and 5, respectively. The high scaling factor allowed for two
layers of elements in the thickness direction for each composite layer without an excessive
number of elements. Once the model was meshed, a non-linear solver was used to achieve
0.02
0.01
0
0.1
0.2
Figure 17. Mesh.
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
0
1651
Resistance Heating for Self-healing Composites
Table 4. Geometry statistics.
(mm)
Composite layers (4)
Length
Width
Height
255
21
0.17
Electrodes (2)
Length
Width
Height
10
12
0.5
Clamps (2)
Length
Width
Height
10
12
10
Table 5. Mesh statistics.
Pre-set resolution
Mesh type
Degrees of freedom
# Mesh elements
# Boundary elements
# Edge elements
Scaling factor
z-resolution of composite
0
Extra fine
Subdomain
81,216
25,329
6673
1172
1 : 1 : 100 (x, y, z)
Two elements per layer
0.1
0
0.2
V
10.45
Figure 18. Potential varies from 0 to 10.45 V at the negative and positive electrodes.
a steady-state solution for both voltage and temperature. The program automatically
iterates until it can converge on a solution that is not time-dependent.
Figure 18 is a plot of potential over the surface of the composite. A large drop in voltage
is observed between the top and bottom surface of the contact interface which is a result
of its very low conductivity and represents contact resistance. A more gradual drop
in voltage occurs along the length of the specimen and again another large drop occurs
on the contact interface layer.
The temperature plot in Figure 19 shows the effects of resistive heating. There is a high
degree of localized heating around the electrodes, a lower area of heating between the
electrodes, and less heating in the end regions outside of the electrodes. The analysis is
revealing in that it allows the temperature directly on the electrode surface to be seen,
whereas in reality this area is obscured by the C-clamp (in the picture the clamp is also
blocking the view of the surface; however the data plots of temperature are recorded
under the clamp). Temperatures in all locations were higher than measured values around
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1652
N. KWOK AND H. T. HAHN
0
0.1
0.2
T(C)
117
237
Figure 19. Temperature varies from 117 to 237 C.
250
Temperature (C)
200
150
100
Contact at 4.0 A
50
FEMLab
Electrode
Electrode
0
0
50
100
150
Position (mm)
200
250
Figure 20. The FEMLab simulation predicts higher heating than measured.
0
0.1
0
0.2
V
10.45
Figure 21. View of electric current lines in the x–y plane.
the electrodes. The plot in Figure 20 compares the measured values to those predicted
from FEMLab.
As shown by the plot, the simulation predicts higher temperatures than observed
experimentally. Higher temperatures near the edges of the composite indicate that thermal
conductivity for this specimen is actually lower than the sample used in the literature.
Overall temperatures are also higher because the actual resistivity is lower than simulated,
indicating that there is some interaction between layers to improve the resistivity of the
middle layers when arranged in a cross-ply layup.
The electric current density is plotted in Figure 21. This illustrates the paths that the
current takes going from the positive to the negative electrode. A close-up of the current
density around the positive electrode (Figure 22) in the x–z plane shows the effects
of anisotropy in the model. Since the conductivity for the middle layers is low in the
x-direction compared to the top and bottom layers, the current lines tend to jump through
the middle layers to get to the bottom layer before traveling in the x-direction.
Thus, most of the current flow is found in the top and bottom layers.
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
Resistance Heating for Self-healing Composites
1653
Figure 22. Close-up view of current lines in the x–z plane reveals the effects of anisotropy.
CONCLUSIONS
The unique properties of carbon fiber composites require special consideration
of contact resistance in any resistive heating application. The two keys to achieving
low contact resistance are removal of any resin rich top layers that may interfere with fiber
contact, and an electrode with a large surface area in pressure contact with exposed fibers.
A challenge lies ahead in reducing contact resistance as much as possible while creating
a removable electrode system that works on the composite surface.
To this end, the C-clamp electrode is a step in the right direction, having demonstrated
to significantly reduce localized heating. However, no matter how good the electrode
connection is, it will still be limited by the resin-rich top layer present in all composites.
Using conventional resins, the only way to overcome this is to remove the top resin layer,
exposing the fibers below. As the data shows, doing so not only greatly reduces contact
resistance, but also improves the consistency of the measurements. Another alternative
to consider is a conducting resin, although these remain largely experimental.
The simulation using FEMLab provides a good approximation of the heating behavior
and the same parameters can be applied in the future to more complex geometries. This
is particularly useful in predicting temperatures in areas unavailable for measurement,
such as underneath the electrode. It is also useful for testing voltages and temperature
levels that would normally be unsafe or damaging to the composite, allowing problems
to be averted without resorting to destructive testing. Most importantly, the results
of these tests are applicable to the self-healing polymer described in the introduction.
The simulation can determine beforehand the required current levels and appropriate
electrode configuration to achieve the desired temperature at a crack location, in order
to effectively heal matrix cracks.
ACKNOWLEDGMENTS
The present paper is based on work supported by the Air Force Office of Scientific
Research through a MURI Grant FA 9550-05-1-0346 to the University of Illinois,
Urbana-Champaign. Appreciation is extended to Dr B. Les Lee for his program
management.
REFERENCES
1. Chen, X., Wudl, F., Mal, A., Shen, H. and Nutt, S. (2003). New Thermally Remendable Highly
Cross-Linked Polymeric Materials, Macromolecules, 36(6): 1802 –1807.
2. Yarlagadda, S. and Kim, H. (2002). A Study on the Induction Heating of Conductive Fiber
Reinforced Composites, Journal of Composite Materials, 36(4): 401–421.
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016
1654
N. KWOK AND H. T. HAHN
3. Abry, J., Bochard, S., Chateauminois, A., Salvia, M. and Girau, G. (1999). In situ Detection
of Damage in CFRP laminates by Electrical Resistance Measurements, Composites Science and
Technology, 59(6): 925–935.
4. Todoroki, A., Tanaka, M. and Shimamura, Y. (2003). High Performance Estimations
of Delamination of Graphite/Epoxy Laminates With Electric Resistance Change Method,
Composites Science and Technology, 63(13): 1911–1920.
5. FEMLab 3.0 Users Manual (2004). COMSOL Inc., Burlington, MA.
6. James, B., Wostenholm, G., Keen, G. and McIvor, S. (1987). Prediction and Measurement of the
Thermal Conductivity of Composite Materials, J. Phys. D: Appl. Phys., 20(3): 261–268.
7. Mills, A. (1999). Basic Heat and Mass Transfer, 2nd edn, Prentice Hall, New Jersey, p. 297.
8. Zoflex Conducting Rubber Datasheet, http://www.irmicrolink.com/zoflexcd45_1.pdf
Downloaded from jcm.sagepub.com at PENNSYLVANIA STATE UNIV on April 8, 2016