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Test MethodA non-contact real-time strain measurement and control system
for multiaxial cyclic/fatigue tests of polymer materials
by digital image correlation method
Gang Tao, Zihui Xia*
Department of Mechanical Engineering, University of Alberta, Edmonton, Alta., Canada T6G 2G8
Received 4 May 2005; accepted 28 June 2005
Abstract
A non-contact real-time strain measurement and control system based on the Digital Image Correlation Method (DICM), has
been successfully established for cyclic/fatigue tests of polymer materials. It allows recording of the evolution of strain during
the entire fatigue life of the polymer materials under test. The stressstrain hysteresis loops can also be accurately recorded
through synchronizing the stress and strain data. A minimum detectable strain of 0.01% was achieved. In addition, through
simplifying the calculation procedure and optimizing the searching algorithm in DICM, a frequency of 10 Hz for strain data
acquisition was reached. This made it possible to perform strain-range-controlled fatigue tests on the specimens. The reliability
and universality of the system was verified by carrying out different types of multiaxial cyclic/fatigue tests on specimens of an
epoxy polymer. The success of this method would facilitate performing various types of fatigue tests on polymer materials and
would allow gaining better insight and understanding of their fatigue and failure behavior.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Non-contact real-time strain measurement; Digital image correlation method; Multiaxial cyclic/fatigue test; Epoxy polymer;
Viscoelasticity; Ratcheting strain; Stress relaxation
1. Introduction
Polymers are viscoelastic or viscoplastic materials with
distinct time-dependent deformation behavior which has
been reported in many publications[15]. Relevant factors
include strain recovery, stress relaxation, accumulation of
ratcheting strain (cyclic creep) in stress-controlled cyclic
tests with a mean stress, stress relaxation in strain-controlled
cyclic tests with a mean strain, etc. It can be foreseen that the
control mode, the loading rate and the loading path will have
significant effects on the fatigue behavior of the polymer
materials. Therefore, acquisition of strain and stress data
during the entire fatigue test process is important for
understanding fatigue mechanisms and development of
theoretical models. Strain gauges and extensometers have
served as conventional strain measurement tools in most
mechanical experiments. Their use in cyclic/fatigue tests
could meet some difficulties. For example, strain gauges
could not be used if fatigue life of the gauges was shorter
than that of the material tested. For relatively soft materials
such as polymers, the knives of the extensometer could
cause local damage (even with protective film) and thus the
obtained fatigue life could be much shorter than that of a
specimen with a smooth surface. There have been a large
number of publications on fatigue tests of polymers, e.g. see
Refs.[69],among others. Fatigue tests can be carried out
under load (stress)-controlled or deformation (strain)-
controlled modes. For rubbers and plastics, the latter type
of fatigue test is usually carried out by controlling a certain
global deformation parameter (grip displacement, rotational
angle, etc.)[10,11]. In most cases, the local strain values are
Polymer Testing 24 (2005) 844855
www.elsevier.com/locate/polytest
0142-9418/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.polymertesting.2005.06.013
*Corresponding author. Tel.:C1 780 492 3870; fax:C1 780 492
2200.
E-mail address:[email protected] (Z. Xia).
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Capture the next image(target image).
Start Experiment.
Read subset
data inreferenceimage.
Capture the referenceimage.
Find the displacement ofselected subset at integerpixel level in target image
within estimateddisplacement range.
Calculate bilinearinterpolation coefficient
matrix.
Calculate correlationcoefficient C at point O(0,0)and its adjacent 8 sub-pixels
positions around O: (1,0),(1,1), (0,1), (-1,1), (-1,0), (-1,-
1), (0,-1), (1,-1)
Is C at O theminimum value?
Thedisplacementat sub-pixel
level is foundand record it.
t
Is experiment finished?
No
Move Owhere t
locating athe next psame dire
Yes
Is
th
Yes
No
Fig. 1. Flow chart of searching procedure of DICM used in real-time strain measurement for cyclic/fatigue t
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uandvare in-plane displacements of the subset center along
x and y directions. vu/vx, vu/vy, vv/vx and vv/vy are
displacement gradients.f(xi,yi) andgx
i ;
y
j are gray valuesof each pixel in the reference and target images,
respectively. All pixels in an m!n pixels subset are taken
into the calculation. A minimum value of the correlation
coefficientC, which represents difference between these two
subsets, will be achieved if the real displacements and
displacement gradients,u, v, vu/vx, vu/vy, vv/vxand vv/vy,
are found. To further simplify the calculation, in our
application, the correlation coefficient was modified as:
CZXmiZ1
XnjZ1
jfxi;yjKgxi ;y
jj (3)
Two small marks spaced a certain distance apart alongthe axial direction were made on the cylindrical surface of
the specimens (see Fig. 4c). The mark points areas are
selected as the subsets and the sizes of them are about 15!
15 pixels. Only the displacement of the subset center is
concerned and, due to the small size of the subset and the
uniformity of the strain field in the gauge area, the values of
third and fourth items on the right side of Eq. (2) are much
smaller in comparison to the values of the second item.
Therefore, neglecting derivative items in Eq. (2) has little
effect on the calculation of displacements of the subset
center, i.e. Eq. (2) can be approximated as,
xi ZxiCu yj ZyjCv (4)
This will significantly simplify the calculation procedure
and increase the search speed.To perform real-time processing, an optimized searching
strategy must be utilized. A two step searching strategy,
which is similar to so called coarse-fine search method[26],
was employed in our image analysis system. In the first step,
searching was operated at pixel level. The searching range
was predefined by an estimation of the movement range of
the subsets. Through the first step, displacements of
reference subsets were acquired at integer pixel level. In
the second step, searching operates at sub-pixel level.
Bilinear interpolation was utilized to obtain the gray value at
sub-pixel position. Minimum displacement of 0.05 sub-
pixel can be detected in our system. Then, the values ofu
andv which minimize the correlation coefficient C (Eq. (3))
are the displacements of the reference subset center.
Instead of calculating coefficient C of each position
within the target subset area, a minimum-value-oriented
searching method was employed. With this method,
searching is along the minimum coefficient oriented
direction. Then, only the points along the searching path
are required to be calculated. The searching speed can
increase more than 20 times by using this optimized
searching method.Fig. 1is the flow chart of the searching
procedure. Fig. 2 displays a typical distribution of the
correlation coefficient from the experiment data at sub-pixel
level. A very smooth surface and a unique minimum valueof such a distribution is the prerequisite for the minimum-
value-oriented searching method.
Once the displacements of the two mark points are
obtained, strains in the gauge area can be derived.
Fig. 3(a) and (b) shows the model for strain calculation.
The cross section and side view of the test gauge of the
specimen are displayed at the left and right sides of
Fig. 3(a), respectively. Circles 1 and 2 represent initial
positions of the two marks before deformation and circles
Fig. 2. Correlation coefficient distribution in DICM at sub-pixel
level.
(a) (b)
L
y1
y2
y2
y1
y2- y2
y1- y1
R
xy
x1 x2
1
2
1'
2'
y1v1
x1
y2
x2u1
=
=
v2
u2
=
=
P
P
L+L
Fig. 3. A schematic for strain calculation: (a) cross section and side view of the specimen; (b) sketch of shear strain calculation.
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10 and 20 represent their positions after deformation. R is
the radius of the outer surface of the cross section of thespecimen and Lis the initial axial distance between these
two marks under zero loading. In the case of a biaxial
loading condition, the mark points will move along both
x and y directions as shown in Fig. 3(a). Since the
specimen undergoes a uniform deformation, difference of
initial positions of the mark points will make no
difference in the strain calculation. Then the axial strain
can be expressed as:
3x ZDL
L
Z
u2Ku1
L
Z
Dx1KDx2
L
(5)
The shear strain can be calculated by:
gZ tgK1 D
LCDL (6)
As shown in Fig. 3(b), D is the relative circumfer-
ential displacement of the outer surface between two
cross sections where marks 1 and 2 are located.Assuming initial angles, a and b, of these two marks
according to the projection plane PP are:
aZ cosK1 y1
R
bZ cosK1
y2
R
(7)
After rotation, the angles change to:
aCDaZ cosK1 y1Cv1
R
Z cosK1
y1KDy1
R
bCDbZ cosK1 y2Cv2
R Z cos
K1 y2KDy2
R
(8)
and the relative rotation angle between these two cross
sections is:
DfZDaKDb (9)
Thus, the relative circumferential displacement is:
DZDf$R (10)
Substituting (10) into (6), the shear strain can be
obtained.
The precision ofDLor D is 0.05 pixel, according to (5)
and (6), the precision of the strains depends on the distance
between two marks. For LZ1000 pixels (z25 mm), the
precision of the normal strain will be 0.005% and the
precision of the shear strain will betgK10.005.
Fig. 4. Tubular and solid specimen made of Epon 826/Epi-Cure
Curing Agent 9551: (a) drawing of the solid specimen; (b) drawing
of the tubular specimen; (c) photo of the tubular and solid specimen
with mark points on the surfaces.
Fig. 5. Configuration of the non-contact real-time strain-range-
controlled fatigue test system by using DICM.
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3. Specimens and test system
Epon 826 Resin and Epi-Cure Curing Agent 9551 were
mixed in the ratio of 100:36 (weight) and then cast into steel
tubes. The castings were cured in an oven for 2 h at 50 8C
and 2.5 h at 120 8C subsequently. Afterwards, the specimens
were cooled to room temperature in the oven. Two types of
specimens, solid specimen and thin-walled tubular
specimen, shown inFig. 4with drawings and photo, were
machined by a CNC lathe. All the uniaxial tests of solid
specimens were performed by a modified MTS system,
details of which have been described in [28]. All the
multiaxial tests of tubular specimen were performed by an
in-house-made triaxial fatigue test machine [29], which is
capable of applying axial load, shear load, internal pressure
and external pressure simultaneously. The whole test system
configuration is shown inFig. 5. A Matrox Meteor-II/Multi-
Channel frame grabber installed in computer A and a Sony
XC-HR70 Monochrome CCD camera (8 bit 1024!768)
were dedicated to image acquisition. Computer A was
equipped with an Intel 2.8 GHz CPU and 1 GB memory to
perform image capturing and analyzing. Computer B was in
charge of data acquisition and controlling the test machine.
These two computers were connected by a network cable to
perform real-time data exchange. The load data acquired
from the load transducer by computer B was captured by
computer A at the exact moment the image was taken. Then,
the load and strain data were synchronized. For the strain-
range-controlled fatigue tests, the strain data calculated by
computer A was fed back to computer B and compared with
the predefined strain range. If the strain exceeds the
predefined strain range, the load direction will be reversed.
A constant loading rate of 10 MPa/s (absolute value) was
chosen in all the strain-range-controlled fatigue tests
0 1 2 3 4 5 6 70
10
20
30
40
50
60
70
80
Strain (%)
Stress(MPa)
Extensometer Camera
Fig. 6. Comparison of recorded stressstrain curves by using
traditional extensometer and using the non-contact real-time strain
measurement and control system.
0 0.5 1 1.5 2 2.5 30
10
20
30
40
50
60
70
Strain (%)
Stress(MPa)
0 20 40 60 80 100 1202.5
2.6
2.7
2.8
2.9
Cycle
Strain(%)
0 20 40 60 80 100 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Cycle
Strain(%)
(d)(c)
(a)
0 0.5 1 1.5 2 2.5 30
10
20
30
40
50
60
70
Strain (%)
Stress(MPa) N=1
N=20, 50, 100
(b)
Fig. 7. Experimental data of stress-controlled uniaxial cyclic test with mean stress: (a) stressstrain loops of the first 5 cycles; (b) stressstrain
loops of the 1st, 20th, 50th and 100th cycle; (c) maximum strains of each cycle; (d) minimum strains of each cycle.
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reported in the current paper. All the tests were conducted at
room temperature and under the laboratory environment.
Each test continued until failure of the specimen.
4. Sample test results and verification
4.1. Uniaxial monotonic tensile test
A uniaxial monotonic tensile test was first performed.
The strain was measured simultaneously with both
extensometer and this new non-contact system. Results
from these two methods are compared inFig. 6,from which
one can see good agreement between them. The small
discrepancy at larger strain might result from non-linear
output of the extensometer at high strain levels.
4.2. Stress-controlled uniaxial cyclic test with mean stress
A stress-controlled uniaxial cyclic test was carried out with
stress range of 60 MPa and mean stress of 30 MPa.Fig. 7(a)
and (b) shows the stressstrain loops of the first 5 cycles and
1st, 20th, 50th and 100th cycle. Maximum and minimum
strains of each cycle are plotted in Fig. 7(c) and (d),
respectively. Non-linear behavior of this material is manifest
at higher strain levels. It is noticed from the figures that, with
the increase of cycles, the stressstrain loops tend to be
slimmer and more linear. Ratcheting strain is accumulated
from the very beginning of loading and the ratcheting rate
decreases cycle by cycle. The ratcheting strain rate became
almost zero after the 20th cycle, i.e. an asymptotic stable state
has been reached. Total ratcheting strains at the maximum and
minimum stresses are 0.3% and 0.55%, respectively. Thisspecimen failed at the 109th cycle.
4.3. Strain-range-controlled uniaxial cyclic tests
with mean strain
A strain-range-controlled uniaxial cyclic test was also
carried out with strain range of 3.2% and mean strain of
1.6%.Fig. 8(a) and (b) shows the stressstrain loops of the
first 5 cycles and the 1st, 50th and 238th cycle. Maximum
and minimum stresses of each cycle are plotted inFig. 8(c)
and (d). It can also be noticed that, with the increase of
cycles, the stressstrain loops tend, again, to be slimmer and
more linear. Stress relaxation occurs from the very
beginning of testing and the relaxation rate is decreasing
with increasing cycles. One can also distinguish that after
the 50th cycle the relaxation rate is almost zero and a stable
state has been reached. Total stress relaxations at the
maximum and minimum strains are K2 MPa andK7 MPa,
respectively. This specimen fractured at the 239th cycle.
0 1 2 320
0
20
40
60
80
Strain (%)
Stress(MPa)
0 1 2 320
0
20
40
60
80
Strain (%)
Stress(MPa)
N=1
N=50, 238
0 50 100 150 200 25060
62
64
66
68
70
Cycle
Stress(MPa)
0 50 100 150 200 250
8
6
4
2
0
Cycle
Stress(MPa)
(d)(c)
(a) (b)
Fig. 8. Experimental data of strain-range-controlled uniaxial cyclic test with mean strain: (a) stressstrain loops of the first 5 cycles; (b) stress
strain loops of the 1st, 50th and 238th cycle; (c) maximum stresses of each cycle; (d) minimum stresses of each cycle.
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It is noted that the strain range in the previous stress-
controlled test is about 2.5% (Fig. 7a and b) while in the
current test the strain range is 3.2%. However, the fatigue
life of the latter is even longer than that of the former. This
might be attributed to the different characteristics in the
stress/strain responses of the different load control modes. In
the former, the ratcheting strain could be a detrimentalfactor to the fatigue life, while the stress relaxation in the
latter could be a beneficial factor to the fatigue life of the
materials.
4.4. Stress-controlled pure shear cyclic test with mean stress
A stress-controlled pure shear cyclic test was preformed
with shear stress range of 36 MPa and mean stress of
18 MPa.Fig. 9(a) and (b) shows the stressstrain loops of
the first 5 cycles and 1st, 100th, 1000th and 3000th cycle.
Maximum and minimum shear strains in each cycle are
plotted inFig. 9(c) and (d). Distinct non-linear behavior can
be seen at higher shear strain level. Similar to the axial
loading tests, the stressstrain loop became slimmer and
more linear with increasing cycles. Moreover, ratcheting
shear strain was accumulated from the first cycle and its rate
decreased with increasing cycles. It is also noticeable that
after the 500th cycle the ratcheting rate is almost zero,
which indicates an asymptotical stable stage is reached.
Total ratcheting strains at the maximum and minimum
stresses are 0.6% and 1.2%, respectively.
4.5. Proportional strain-range-controlled biaxial cyclic test
A proportional axial-shear biaxial cyclic test wasperformed under strain-range-controlled mode with axial
strain range between 0 and 2.06%. The shear loading rate
was kept proportional to the axial one in this test. From the
experimental results we noticed that, although only axial
strain range was controlled, the shear strain was also kept
within a constant range during the entire test process.
Therefore, proportional strain-range-controlled biaxial cyc-
lic tests can be successfully performed by constraining the
strain range in only one direction.Fig. 10(a) and (b) shows
the stressstrain loops of the first 5 cycles and 50th, 1000th
and 1700th cycles in the axial and shear directions,
respectively. Fig. 10(c) and (d) shows the mean stresses of
each cycle in the axial and shear directions, respectively.
Non-linear behavior can be noticed in both directions and
both the axial and shear stressstrain loops become slimmer
and more linear with increasing cycles. Stress relaxation
occurs in both directions and the rates decrease with
increasing cycles. Total mean stress relaxations are 4.0 MPa
and 2.2 MPa in axial and shear direction, respectively. This
specimen failed at the 1716th cycle.
0 1 2 3 4 50
10
20
30
40
Strain (%)
ShearStress(MPa
)
0 1 2 3 4 50
10
20
30
40
Shear Strain (%)
ShearStress(MPa) N=1
N=100
N=3000
N=1000
0 500 1000 1500 2000 2500 3000 35004.3
4.4
4.5
4.6
4.7
4.8
4.9
5
Cycle
ShearStrain(%)
0 500 1000 1500 2000 2500 3000 35000
0.2
0.4
0.6
0.8
1
1.2
Cycle
ShearStrain(%)
(d)(c)
(a) (b)
Fig. 9. Experimental data of stress-controlled pure shear cyclic test with mean stress: (a) stressstrain loops of the first 5 cycles; (b) stressstrain
loops of the 1st, 100th, 1000th and 3000th cycle; (c) maximum shear strains of each cycle; (d) minimum shear strains of each cycle.
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4.6. Non-proportional stress-controlled biaxial cyclic test
A non-proportional biaxial test was also carried out
under stress-controlled mode with axial stress range of
050 MPa and shear stress range of 028 MPa.Fig. 11(a)
shows the quarter-circle sectorial loading path of this test.
Fig. 11(b) shows the fan-shaped strain response of the first
10 cycles and 50th, 200th and 700th cycles of this test. Mean
strains of each cycle in the axial and shear directions are
plotted inFig. 11(c) and (d), respectively. It is also noticed
that ratcheting strains are accumulated in both axial and
shear directions. Total ratcheting mean strains are 0.38%
and 0.44% in axial and shear directions, respectively. Afterthe 400th cycle the ratcheting rates become zero and
material response is in a stable stage. After the 550th cycle,
ratcheting rates are seen to increase again until final failure.
Such phenomenon is more pronounced in the axial
direction. This specimen failed at the 716th cycle.
5. Discussions
Due to the required time for digital image processing, the
sampling frequency of the current system is not as high as
that by using the traditional strain gauge or extensometer.
Due to the transferring of the image from CCD to frame
grabber and the delay time resulting from the image
processing, the feedback strain signal would be delayed
about 0.1 s. During this period the machine actuator would
keep increasing the load until the computer B received the
strain reading and commanded the machine to reverse the
load direction. Therefore, the actual strain peak values
(maximum and minimum strains) that the specimen was
subjected to, would be larger than the predefined limits.
Because of the influence of several other factors, such as
sampling frequency and the loading rate, the actual strain
limit will slightly fluctuate around its mean value from cycle
to cycle. The sampling frequency is determined by the
image processing time. In our system, the average
processing time for two 15*15 pixels subsets is 0.05 s. To
synchronize with the CCD camera, the interval between two
sample points is 0.1 s. Then the maximum sampling
frequency of this system is 10 Hz. Theoretically, increase
of sampling frequency will enhance the precision of
capturing the peak strains and, therefore, suppress the
fluctuation of the measured strain limit values. The degree
of the fluctuation is also determined by the loading rate.
Higher loading rate will intensify the fluctuation because
0 0.5 1 1.5 2 2.510
0
10
20
30
40
50
60
Axial Strain (%)
AxialStre
ss(MPa) N=1-5
N=50, 1000, 1700
0 1 2 3 410
0
10
20
30
40
Shear Strain (%)
ShearStress(MPa) N=1-5
N=50, 1000, 1700
0 500 1000 1500 200020
21
22
23
24
25
26
Cycle
AxialMeanStress(MPa)
0 500 1000 1500 200012
12.5
13
13.5
14
14.5
15
Cycle
ShearMeanStress(MPa)
(d)(c)
(a) (b)
Fig. 10. Experimental data of proportional strain-range-controlled biaxial cyclic test under combined axial-shear loading: (a) axial stressstrain
loops of the first 5 cycles and the 50th, 1000th and 1700th cycle; (b) shear stressstrain loops of the first 5 cycles and the 50th, 1000th and 1700th
cycle; (c) axial mean stress in each cycle; (d) shear mean stress in each cycle.
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the strain increment will be larger than that under lower
loading rate within the same data sampling interval.
Moreover, the inertia of the actuator will also lead to such
fluctuation.Fig. 12(a) and (b) shows the histogram of the
deviation of the maximum and minimum strain of each
cycle to their mean value in the 03.2% strain-range-
controlled fatigue test, respectively. The loading rate is
10 MPa/s, which was chosen as the unified loading rate for
each strain-range-controlled uniaxial fatigue test in the
current study. The mean value and standard deviation of the
maximum strain of all cycles in the test are 3.2 and 0.026%,
respectively. The mean value and standard deviation of the
minimum strain of all cycles are 0 and 0.014%, respectively.
The mean value and standard deviation of minimum strain
10 0 10 20 30 40 50 605
0
5
10
15
20
25
30
Axial Stress (MPa)
ShearStress
(MPa)
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
Axial Strain (%)
ShearStrain(%) N=1-10
N=50, 200, 700
0 200 400 600 8001.7
1.9
2.1
2.3
2.5
Cycle
AxialMeanStrain(%)
0 200 400 600 8003
3.2
3.4
3.6
3.8
Cycle
ShearMeanStrain(%)
(d)(c)
(a) (b)
Fig. 11. Experimental data of non-proportional stress-controlled biaxial cyclic test under combined axial-shear loading: (a) biaxial loading path;
(b) biaxial strain response of the first 10 cycles and the 50th, 200th and 700th cycle; (c) axial mean strain in each cycle; (d) shear mean strain ineach cycle.
0.06 0.04 0.02 0 0.02 0.04 0.060
5
10
15
20
25
30
Strain Deviation (%)
NumberofCycles
0.04 0.02 0 0.02 0.040
10
20
30
40
50
Strain Deviation (%)
NumberofCycles
(a) (b)
Fig. 12. Histograms of the deviation of the strain range: (a) the deviation of the maximum strain; (b) the deviation of the minimum strain.
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are slightly lower than that of the maximum strain. This can
be explained by this kind of epoxy material having lower
stiffness at high stress level. Therefore, the strain rate is
higher at higher stress level than that at lower stress level
under the same absolute loading rate. According to
Fig. 12(a), the maximum error in the peak strain is about
0.06%, and then the relative maximum error of the strainrange is about 1.9%. Such small error should have very little
effect on the fatigue life of the specimen.
6. Conclusions
A non-contact real-time strain measurement and
control system based on the digital image correlation
technique has been established. The method is non-
destructive and has no upper limitation for the strain
measurement. Therefore, it can be used in multiaxialfatigue tests of soft materials such as polymers.
Currently, the system can measure strains with an
accuracy of 0.01% and a frequency of 10 Hz for strain
data acquisition (or 0.1 s for retrieval of the measured
strain value). As such, the system even allows running
strain-range-controlled fatigue tests with moderate load-
ing rates. The capability of the system has been verified
through recording the stressstrain responses of various
types of uniaxial and biaxial cyclic/fatigue tests of an
epoxy polymer material. The results show that the
evolution of stressstrain hysteresis loops during the
entire fatigue life of the specimen can be accurately
recorded. The different responses in the stress and the
strain-range-controlled tests also indicate the significant
effect of the loading modes on the fatigue behavior and
fatigue life of viscoelastic/viscoplastic and time-depen-
dent materials such as polymers. The success of this
method would facilitate performing various types of
fatigue tests on polymer materials and would allow us to
gain more insight and understanding of the fatigue and
failure behavior of polymer materials. This system has
potential for further upgrade in the future. Utilization of
a higher resolution CCD camera will improve the strain
detection precision. It will be possible to further reduce
the processing time and to increase the strain dataacquisition frequency if the computer and the CCD
camera used possess higher speeds than the current ones.
Acknowledgements
The work presented here is part of a general investigation
of the mechanical properties and damage of advanced
composite materials. The research is supported by the
Natural Sciences and Engineering Research Council of
Canada (NSERC) through grant to Z.X.
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