Progress Report 3—BSEE Contract E12PC00033 (Legacy No. M11PC00029) The Effect of Deformation Damage on the Mechanical Behavior of Sea Ice: Ductile-to-Brittle Transition, Elastic Modulus, and Brittle Compressive Strength Erland M. Schulson, PI; Carl E. Renshaw, Co-PI; Scott A. Snyder, Ph.D. candidate Ice Research Laboratory Thayer School of Engineering, Dartmouth College, Hanover, NH 18 September, 2014 1 Introduction In this third annual progress report, we summarize the results accumulated to date from our experiments studying the effects of damage on the elastic properties and mechanical behavior of ice. To review, the damage we have been studying is a result of prior inelastic strain, which was imparted into specimens of freshwater ice and saline ice, under compression at strain rates in the ductile regime. Both types of ice were produced and tested in the laboratory. The tests were designed to address the effects of compressive prestrain on three specific aspects of ice: (i) Elastic modulus. (ii) Ductile-to-brittle transition. (iii) Brittle compressive strength. We anticipate this research to have impact in improving the safety and design of offshore arctic structures. The results will broaden our understanding of ice to include— in addition to the virgin material—that which has a mechanical history closer to what may be encountered in the field. Does a history of prior deformation affect the way ice responds upon impact with a structure? Ice can be ductile when compressed slowly, but brittle when compressed rapidly, above a certain critical rate. This ductile-to-brittle transition strain rate is often where the maximum strength of ice occurs. How does prior strain affect the critical transition rate, and therefore the behavior and strength of ice? In the process of addressing such questions, we have also developed quantitative ways to measure damage and to distinguish ductile and brittle behaviors in the bulk specimens. The next sections report these methods and the results of our work. 1
The Effect of Deformation Damage on the Mechanical Behavior of Sea
Ice: Ductile-to-Brittle Transition, Elastic Modulus, and Brittle.
Compressive StrengthProgress Report 3—BSEE Contract E12PC00033
(Legacy No. M11PC00029)
The Effect of Deformation Damage on the Mechanical Behavior of Sea
Ice: Ductile-to-Brittle Transition, Elastic Modulus, and
Brittle
Compressive Strength
Erland M. Schulson, PI; Carl E. Renshaw, Co-PI; Scott A. Snyder,
Ph.D. candidate
Ice Research Laboratory
18 September, 2014
1 Introduction In this third annual progress report, we summarize
the results accumulated to date from
our experiments studying the effects of damage on the elastic
properties and mechanical
behavior of ice. To review, the damage we have been studying is a
result of prior inelastic
strain, which was imparted into specimens of freshwater ice and
saline ice, under
compression at strain rates in the ductile regime. Both types of
ice were produced and
tested in the laboratory. The tests were designed to address the
effects of compressive
prestrain on three specific aspects of ice:
(i) Elastic modulus.
(ii) Ductile-to-brittle transition.
(iii) Brittle compressive strength.
We anticipate this research to have impact in improving the safety
and design of
offshore arctic structures. The results will broaden our
understanding of ice to include—
in addition to the virgin material—that which has a mechanical
history closer to what
may be encountered in the field. Does a history of prior
deformation affect the way ice
responds upon impact with a structure? Ice can be ductile when
compressed slowly,
but brittle when compressed rapidly, above a certain critical rate.
This ductile-to-brittle
transition strain rate is often where the maximum strength of ice
occurs. How does prior
strain affect the critical transition rate, and therefore the
behavior and strength of ice? In
the process of addressing such questions, we have also developed
quantitative ways to
measure damage and to distinguish ductile and brittle behaviors in
the bulk specimens.
The next sections report these methods and the results of our
work.
1
Step Procedure
2 Prestrain at constant strain rate, uniaxial compression.
3 Measure bulk elastic properties [E, G,ν ], mass density [ρ] and
porosity [φ ].
4 Examine microstructure and quantify damage:
measure crack density [scalar, ρc] and [tensor, α], and
recrystallized area fraction [ frx].
5 Reload to obtain σ -ε curves under uniaxial compression.
2 Experimental procedures We have continued to follow the same
experimental procedures described in our previous
Progress Reports and outlined in Table 1 above. These procedures
involve growing and
preparing specimens of columnar ice, loading the specimens in
uniaxial compression at
a constant strain rate εp to impart prestrain damage, obtaining
elastic properties using
ultrasonic transmission techniques, and then reloading the
specimens—again under
uniaxial compression—to observe how their behavior may have changed
as a result of
damage. All tests were run at (−10.0±0.2) C. Figure 1 shows a
diagram of (a) the
prestrain loading and (b) the configuration of rectangular prism
subspecimens cut from
the prestrained parent specimen.
(b) subspecimens (a) parent specimen
Figure 1: Typical geometry (a) with respect to uniaxial loading by
across-column compression
along x1 to impart prestrain, εp, in parent specimen, which yields
(b) two pairs of subspecimens
oriented in either the x1 or x2 direction. Parent specimens were
machined to cubes of 152 mm
sides; subspecimens, to 60 mm × 60 mm × 120 mm.
2
To explore the effect of prestrain rate, we compressed specimens at
both one and
two orders of magnitude below the inherent ductile-to-brittle
transition strain rate,
εD/B,0, for undamaged material. At −10 C this rate for freshwater
columnar ice is about
1×10−4 s −1, ten times lower than εD/B,0 for saline ice at 1×10−3 s
−1. Table 2 lists for
both types of ice the levels of εp at each εp that have been tested
to date. Photographs
in Figure 2 show the progression from the undamaged state through
specified levels of
prestrain in parent specimens of both freshwater ice and saline
ice.
Table 2: Uniaxial compressive prestrain conditions tested for (F)
freshwater ice and (S) saline ice.
Prestrain rate, εp Prestrain level, εp
(s −1) 0.003 0.035 0.085 0.100 0.15 0.20
1×10−6 F F F F - -
1×10−5 F, S F, S - F, S F F
1×10−4 - S - S - -
(a) (b) (c) (d)
undamaged 0.003 0.035 0.10
(e) (f) (g) (h)
undamaged 0.003 0.035 0.10
Figure 2: Specimens of freshwater ice (top row) and saline ice
(bottom row) after specified levels
of prestrain εp, indicated below each photograph. Prestrain was
imparted in the x1 direction −1(vertical in these images) at a
constant strain rate of 1×10−5 s .
3
The mass density ρ of the specimen was recorded before and after
prestraining and
re-milling. ρ was calculated by weighing the specimen and dividing
by its volume as
the product of measured lengths in each direction. The bounding
volume included any
porous space of cracks imparted through prestraining. Porosity was
calculated as
(ρ0 −ρ) φ = (1)
ρ0
where ρ0 = 917.45kg m−3 is the expected density of damage-free ice
at −10 C.
After making ultrasonic measurements (by our previously described
method; see
also Section 3.2.2 below), we individually reloaded the
subspecimens at a constant strain
rate εr, ranging from 10−5 to 10−2 s −1, compressing uniaxially in
the long dimension.
3 Results and Discussion This section presents an updated sampling
of the experimental results and discusses their
significance.
Figure 3: Stress-strain curves of virgin freshwater ice (blue) and
of virgin saline ice (green) at
−10 C during compression to 0.10 prestrain applied at the strain
rates εp indicated.
3.1 Prestrain Figure 3 plots typical stress-strain curves recorded
during prestrain. The curves display
the characteristics of ice under ductile compression at constant
strain rate, with a peak
stress typically between strains of 0.002 and 0.005 followed by
softening until about
0.04 strain, after which a steady state was approached. The peak
stress increased with
prestrain rate in each material. Prestrain caused recrystallization
in addition to cracking.
Both of these changes were quantified by inspecting thin sections,
examples of which
appear in Figure 4.
1 cm
(c) (f)
Figure 4: Thin sections of freshwater ice after (left column) 0.035
and (right column) 0.100 −1prestrain at 1×10−5 s at −10 C. Sections
were taken normal to x1. Crossed-polarized light
revealed the grain structure in (a) which contained relatively
little recrystallization, and in (d)
where more recrystallization was evident. The cracks evident under
scattered light in (b) and (e)
were digitally traced to produce the fracture patterns shown in (c)
and (f). Orthogonal vectors
(in red) show the principal directions of the crack density tensor
α (see Section 3.3, Equation 6),
each scaled to the Young’s modulus derived from the corresponding
component of α .
5
3.2 Elastic properties In both types of ice at −10 C, the mass
density decreased and porosity increased with
increasing prestrain, shown in Figure 5. Porosity φ was calculated
using Equation 1. Due
to the presence of brine pockets and pores, even in undamaged
saline ice φ was typically
measured around 1 % to 2 % porosity with some samples perhaps
containing larger
brine channels as high as 6 % to 7 % porosity. The porosity of
undamaged freshwater ice
always measured very near zero. Despite this difference in the two
types of ice, the linear
trends relating φ to εp at corresponding prestrain rates are
remarkably similar for both,
although the εp values are shifted higher in saline ice by one
order of magnitude.
Figure 6 graphs Young’s modulus E versus εp. E decreased with
increasing levels
and rates of prestrain ; similar trends were seen in shear modulus
G and bulk modulus K.
Likewise, the velocities of both P– and S–waves decreased with εp
and with εp, implying
a reduction in stiffness of damaged ice that is not merely due to
its lower mass density.
Furthermore, mass density and Young’s modulus were reduced by a
greater amount when
compression occurred at the higher strain rate in either type of
ice, an indication that the
effects of damage are more pronounced the closer εp is to εD/B, 0.
We measured little to
no detectable effect of damage on Poisson’s ratio, ν .
3.2.1 Prestrain-induced anisotropy
Elastic moduli differed depending on whether prestrained ice was
measured in a
direction either parallel (x1) or perpendicular (x2) to initial
loading. The same level of
strain imparted along the x1 direction tended to cause a greater
reduction in E measured
along x2, ranging from slightly more than up to twice as much as
that measured along x1.
To our knowledge, such prestrain-induced anisotropy has not
previously been reported in
ice.
When using the ultrasonic transmission technique (described in
Progress Reports 1
and 2) to measure the elastic moduli of the prestrained ice,
specimens were loaded
under a typical force of 0.4 kN (corresponding to a compressive
stress of 0.1 MPa) in the
direction of the transmitted pulse. In the course of this study,
the question arose as to
whether the measured elastic moduli are sensitive to the load
applied on the specimen at
the time of measurement. This question has particular relevance in
the presence of what
appears to be, as we observed, a damage-induced anisotropy (i.e.,
greater compliance in
the transverse x2 direction compared to the longitudinal x1
direction). If such anisotropy
were due to cracks aligning preferentially parallel to the
direction of prestrain, would we
find the anisotropy to diminish upon closure of those transverse
cracks under sufficient
6
(a)
(b)
Figure 5: Porosity</> of columnar (a) freshwater ice and (b)
saline ice measured at - 10 °C as a function of prestrain. Shaded
zones indicate 95 % confidence intervals about the linear fits,
weighted for heteroscedasticity.
7
(a)
(b)
Figure 6: Young’s modulus of columnar (a) freshwater ice and (b)
saline ice measured along
one of the two across-column directions (x1 or x2 as indicated
above each panel) at −10 C as a
function of prestrain applied by uniaxial compression in x1. Lines
connect mean values for each
prestrain group and errorbars indicate 95 % confidence intervals
about the means.
8
compression? We tested this idea by increasing the load on the
specimen slowly, at
10 N s−1, and holding at 1 kN, 2 kN, 3 kN, and 4 kN to take
ultrasonic readings. Upon
reaching 4 kN, or about 1.1 MPa, the specimen could not support the
load for more than a
few minutes before fracturing. We found no significant difference
in the elastic moduli
measured at any of the stresses from 0.1 MPa to 1.1 MPa in either
direction (x1or x2).
Although these tests did not support the hypothesis that crack
closure would reduce
the anisotropy, they do not rule it out conclusively, given that we
observed additional
cracks nucleating during the process of applying the higher loads
(> 1 kN). However,
these results suggest that other factors—perhaps the development of
a crystallographic
texture through dynamic recrystallization—in addition to cracks are
responsible for this
prestrain-induced anisotropy, which persists even after further
compression.
The nature of the observed damage-induced anisotropy warrants
further study.
3.2.3 Young’s modulus versus porosity
The relationship between Young’s modulus E and porosity φ is
illustrated in Figure 7.
In these graphs, data from individual subspecimens tested at all
levels of prestrain are
plotted together regardless of prestrain rate. The colors indexed
in the legend indicate
the level of prestrain εp, where ‘0’ refers to undamaged, as-grown
specimens. Young’s
modulus appears to decrease linearly with porosity over the range
of conditions we have
tested, represented by a relationship such as
E = E0 + mφ (2)
where E0 refers to the Young’s modulus of undamaged ice of zero
porosity. Table 3 lists
values for the slope m calculated from a least squares regression
for each type of ice
and in each direction, x1 or x2. For E measured in the x1
direction, the trends are nearly
indistinguishable between saline ice (Fig. 7a) and freshwater ice
(Fig. 7b) up to 10 %
porosity, although the former data show greater scatter. In the x2
direction, compared to
x1, the slope of the fitted line drops slightly to −0.46 from −0.36
in saline ice (Fig. 7c)
and more substantially to −0.63 from −0.35 in freshwater ice (Fig.
7d), again showing
evidence of prestrain-induced anisotropy.
Figure 8 combines the means of x1 measurements from common parent
specimens
(i.e., each of the data here represents the average over two to
four subspecimens cut from
the same initial block of prestrained ice; see Fig. 1) of both
freshwater and saline ice with
an aggregate trend line, along with previously reported
measurements of sea ice from the
sources noted in the legend. The previous data are shown twice:
First, in gray, the values
originally published by Langleben and Pounder (1963) who, lacking
S-wave velocity
measurements, calculated E by assuming a value for Poisson’s ratio
of ν = 0.295 based
9
Figure 7: Youngs modulus E versus porosity φ in freshwater ice and
saline ice, at −10 C, after
the level of prestrain indicated in the legend. E was measured
along x1 (top row) or along x2 (bot
tom row) in the two materials. Shaded zones indicate 95 %
confidence intervals about the linear
fits, excluding in (b) data for φ > 10% and in (a) and (c) data
from the anomalous damage-free
saline ice with φ > 6%.
Table 3: Values of the slope m of the linear regression predicting
Young’s modulus as a function
of porosity (Eq. 2, Fig. 7). Young’s modulus was measured either
parallel (x1) or perpendicular
(x2) to the direction of prestrain applied to the type of ice as
indicated. Slope values are in units of
GPa / % porosity.
10
Figure 8: Youngs modulus versus porosity. Data from current work
(circles) are compared with
previous field data (squares) from Langleben and Pounder
(1963).
11
on other tests. Except for the highest prestrain cases, our data
generally follow the same
trend but slightly below the original field data. The equations
giving elastic moduli in
terms of ultrasonic velocities can be solved for the S-wave
velocity cS in terms of P-wave
velocity cP and Poisson’s ratio:
1−2ν c 2 = c 2 (3)S P 2(1−ν)
This allows Young’s modulus to be written as
(1+ ν)(1−2ν) E = ρcP
2 (4) (1−ν)
Using this formulation, the original Young’s modulus values were
rescaled using an
alternative analysis of the dynamic Poisson’s ratio for sea ice
(Timco and Weeks, 2010),
which at −10 C gives ν = 0.34. The adjusted values are shown in red
in Figure 8.
The linear fits through the original and adjusted sea ice data
closely match the slope
of our laboratory data and bound them on either side. The field
measurements were
made from vertical ice cores, placing transducers at the ends for
ultrasonic transmission
along their lengths, which corresponds to what we have defined as
the x3 direction. This
along-column direction is marginally stiffer than across-column
directions, possibly a
minor factor contributing to slightly higher moduli in the original
field data.
3.3 Damage Porosity (Eq. 1) provided one measure of damage. We also
assessed damage by
recording acoustic emissions during prestraining and by inspection
of thin sections
(e.g., Figs. 4b and 4e), tracing individual cracks (e.g., Figs. 4c
and 4f). As of last year’s
Progress Report 2, we had quantified damage only in terms of the
scalar crack density ρc,
calculated by averaging the squares of crack half-lengths ci over
the visible thin section
area A, typically 50 cm2:
ρc = ∑ci 2/A (5)
This dimensionless damage parameter assumes crack lengths are
uniformly represented
across all crack orientations. In order to describe the
orientations of cracks as well as
their spatial extent, we have borrowed the concept of a crack
density tensor (developed
by Kachanov (1980)), which is given in two dimensions as
1 α = ∑(c 2 n n)i (6)
A i
12
where n is a unit vector normal to the ith crack trace of length
2c. In the summation, nn denotes the dyadic, or outer product,
yielding a second-rank tensor. α is a generalization
of the damage parameter that simplifies to the scalar ρc when crack
orientations are
isotropic or random.
3.3.1 Comparison with non-interacting crack model
With a quantification of damage in terms of crack density, we
tested a non-interacting
crack model (based on continuum damage mechanics) against our
elastic modulus data.
In choosing this model, we assume that cracks do not interact
significantly at prestrain
levels of most practical interest (εp ≤ 0.10, before cracks begin
opening substantially).
The model does not require the absence of interactions altogether,
but rather that stress
amplifications and stress shielding effects mutually cancel each
other.
For the three-dimensional case of randomly oriented cracks, in
which α simplifies to
α11 = α22 = α33 = ρc/3, Kachanov (1992) derived the effective
Young’s modulus
−1 16(1−ν0
9(1−ν0/2)
as a result of damage measured by ρc and in terms of E0 and ν0, the
Young’s modulus
and Poisson’s ratio, respectively, of the corresponding undamaged
material. In the two
dimensional case, again for randomly oriented cracks, α is
isotropic and the effective
Young’s modulus simplifies to
]−1 Eeff, 2D = E0 [1+ πρc (8)
Figure 9 plots Eeff predicted according to Equations 7 and 8 with
the values obtained
by ultrasonic transmission for Young’s modulus Ei (measured in the
xi direction) of the
prestrained specimens of freshwater ice as a function of crack
density component αii.
The 2D and 3D models provide lower and upper bounds, respectively,
for most of the
data. The theoretical (solid) curve of Eeff, 2D closely follows the
trend in experimental
values of E1 up to 0.10 prestrain, in the range where crack density
components in the x1
direction were small. At greater crack densities, the 2D model
underpredicts Young’s
modulus.
The discrepancies may be explained by a host of factors, including
the limitations
of our experimental instruments. The resonant frequency of the
ultrasonic transducers
(200 kHz) implies an upper bound on the length of detectable
cracks. In undamaged
and lightly prestrained ice, P-wave velocities were near 3.8 km
s−1. For this value of cP
13
Figure 9: Comparison of measured and theoretical Young’s modulus of
damaged freshwater ice
as a function of dimensionless crack density component. Theoretical
values (of Eeff, 2D, solid
curve; and of Eeff, 3D, dotted curve) assume non-interacting
cracks.
the corresponding wavelength for our system is 19 mm, thus cracks
with half-lengths
> 1 cm may not contribute to the measured elastic properties.
Another factor to consider
is the observed change in the nature of damage, namely the
preponderance of large
cracks opening as wide as 2 cm with strain levels above 0.10, and
therefore the difficulty
of measuring representative intact subspecimens. These higher
levels of damage may
exceed either the valid range in which cracks can be assumed to be
non-interacting,
or—perhaps more fundamentally—the range in which the material can
be considered as
a continuum.
3.4 Recrystallization The thin sections photographed under
cross-polarized light revealed greater recrystal
lization with increasing levels of prestrain. The area fraction frx
of recrystallized grains
relative to the thin section area A was measured and is graphed in
Figure 10 as a function
of prestrain. The uncertainty in recrystallized area fraction
indicated by vertical error
bars on the graphs for both types of ice was estimated to be ±0.05
based on the average
14
(a)
1.00
(b)
strain rate (s- 1 ) --.
Figure 10: Recrystallized area fraction frx versus prestrain in (a)
saline ice and (b) freshwater ice. The freshwater ice data for each
Sp were fitted with an Avrami-type function (Eq. 9).
discrepancy between counts of the same thin sections made by two
separate researchers. Although the data have some scatter that
increases with prestrain, the trend appears in both types of ice
that less recrystallization occurs for the same cp when imparted at
the higher strain rate. This inverse relationship of frx to £p
suggests that the kinetics of recrystallization depend on time at
the scale of these compression tests. For example, shortening by 10
% at 1 x 10- 6 s- 1 requires over 30 hours . .In freshwater ice,
this was sufficient time to allow 50 % to 90 % of the area to
recrystallize, compared to only 25 % to 50 % when the same
prestrain was imparted in just 3 hours, i.e., 10 times faster
at
11 x 10- 5 s- . The frx data from freshwater ice for each prestrain
rate were fit with an A vrami-type function as follows
(9)
The curves of these relationships are also graphed in Figure lO(b).
Interestingly, recrystallization was substantially greater in
saline ice compressed at the same strain rate. At 0.035 and 0.100
prestrain, frx measured in saline ice at both rates £p tested was
comparable to that in freshwater ice at Sp respectively one order
of magnitude lower.
15
(i) visual
(iii) strain energy (via integration of σ -ε curve)
We developed approaches (ii) and (iii) because, as we discussed in
Progress Report 2, a
visual approach (i) based on the bulk specimen appearance turns out
to be inadequate.
A more consistent characterization of ductile versus brittle
behavior can be made by
examining the stress-strain curves, examples of which are arrayed
in Figures 11 and 12
for freshwater ice and saline ice, respectively, after prestrain of
εp = 0.035. The plots are
paired for each material with reloading in x1 on top and reloading
in x2 on the bottom.
Tests run under the same conditions are overlayed to demonstrate
the reproducibility
of the behavior, which was usually very close except for variations
in post-peak-stress
softening among specimens tested near the transition.
Examination of the σ -ε curves allowed us to identify brittle
behavior, marked by a
sharp peak, σmax, followed by an abrupt drop in axial stress, −Δσ ,
as seen at high εr .
We followed the same criteria explained in Progress Report 2,
defining the macroscopic
mechanical behavior quantitatively as:
Brittle– when −Δσ > 0.5σmax within Δε ≤ 0.001 strain after σmax
occurs, and
Ductile– otherwise, i.e., when σ remains > 0.5σmax beyond 0.001
strain after peak.
Figure 13 charts (for (a) freshwater ice and (b) saline ice) our
characterizations of
mechanical behavior—ductile versus brittle as defined above—for the
prestrain and (x1)
reloading conditions tested.
As another way of discerning the transition strain rate, we
integrated under the
stress-strain curves to calculate a strain energy density, u.
Figure 14 shows an example
for saline ice with (a) the stress-strain curves overlayed for
reference beside (b) the
graphs of u versus strain. The specimen reloaded at 1×10−2 s −1 was
representative
of all brittle specimens of both types of ice, in that it never
developed a strain energy
density greater than 5 kJ m−3. In distinct contrast, u reached over
an order of magnitude
greater value even in the specimen reloaded at 3×10−3 s −1, which
suffered complete
loss of strength just after additional strain of 0.06. Photographs
keyed to the right of the
figure were taken of the subspecimens at the end of reloading. This
strain energy analysis
supported the characterizations of ductile and brittle behavior
charted in Figure 13.
16
6
5
(strain rate/ s·') 0.1
0.02 strain s, after 0.035 prestrain in x, at 10.s s·'
6
5
2 after 0.035 prestrain in x, at 10·5 s·'
(b)
(a)
Figure 1I: Stress-strain curves by strain rate for freshwater ice
reloaded (a) in x 1 and (b) in x2
after 0.035 prestrain imparted at l X Io-5 s- I in X1 . Arrows mark
the ductile-to-brittle transition rate.
17
6
5
-3
0.1
1 at 10"5 s·'
10 (strain rate/ s·'>
002 strain c after 0.035 prestrain in x, at 10·5 s·'
0.1 log
2
(a)
(b)
Figure 12: Stress-strain curves by strain rate for saline ice
reloaded (a) in x 1 and (b) in x2 after 0.035 prestrain irnpatted
at 1 x 10- 5 s- 1 in X1 . An-ows mark the ductile-to-brittle
transition rate.
18
~ 0 1 °'3 ~
Di 0 1
0.003 L 0.003
0 O.Q35 0.10 0.15 0.20 0 0.035 0.10 0.15 0.20 prestrain(a)
prestrain (b)
X1~2 ~ °'1 o1I 1e- 03 (/)
<li ..J2 ro
·~ 02 o, o,
o, 1e-06 L0.003
i 1e- 04 s-1
mode X B o D
mode X B
D D
Figure 13: Mechanical behavior-denoted as ductile (011 ) or b1ittle
(x 11 ) , where n is the number of tests under given conditions- of
(a) freshwater ice and (b) saline ice at - 10 °C after various
levels of prestrain (plotted on the horizontal axes) and upon
reloading in x1 at the strain rate plotted on the vertical
axes.
19
(a) (b)
Figure 14: (a) Stress-strain curves, which were integrated to graph
(b) strain energy density u as
a function of strain, for three subspecimens of saline ice
prestrained to 10 % at 1×10−5 s −1 and
reloaded at different rates noted beside each curve.
The ductile-to-brittle transition appeared more sensitive to
prestrain when that
prestrain was imparted at a rate εp closer in magnitude to the
inherent transition strain
rate, εD/B,0, of undamaged material. This trend occurred for both
freshwater ice and
saline ice. To understand this trend, we refer back to the
quantification of porosity due
to cracking (Fig. 5) and of recrystallization (Fig. 10). Both
processes—cracking and
dynamic recrystallization—act to relieve internal stresses, but the
two seem to compete
at different time scales. The factor that appears to have more
influence on the observed
effects of prestrain is the accumulation of non-propagating cracks.
Freshwater ice that
was prestrained, for instance, by 3.5 % at 10−5 s −1 appeared to
contain relatively few
recrystalized grains but numerous cracks (Figs. 4a and 4b)—its
elastic moduli were
reduced by ∼ 5 % in x1 (Fig. 6a) and εD/B was increased by a factor
of 3 to 10 (Fig. 13a).
The effects of damage were comparable in both materials when each
was prestrained at a
rate the same relative order of magnitude below its inherent
undamaged transition strain
rate.
20
3.5.1 Creep-versus-fracture model
A question that remains is: How well do theoretical models explain
our observations of
prestrain effects on transition strain rate? Progress Report 2
included a description of the
model developed by Renshaw and Schulson (2001), which we restate
here for reference.
This model expresses the micromechanical competition between two
processes: the
intensification and the relaxation of internal stresses at crack
tips. Crack propagation
dominates in brittle fracture, whereas crack blunting dominates in
ductile creep. The
transition strain rate shifts depending on which of the two
mechanisms wins out. The
resistance of a material to crack propagation can be measured by
its plane-strain fracture
toughness, KIc. Creep behavior is modeled as secondary creep which
follows a power
law relationship, ε = Bσn, the parameters of which can be
experimentally determined.
Under uniaxial compression, i.e., zero confinement, the model
predicts the critical
ductile-to-brittle transition strain rate according to
(n + 1)2(3) n−
BKn Icεtc = √ (10)
n/2n π(1−µ)c
in terms of the power-law creep parameters B and n, fracture
toughness KIc, coefficient of
kinetic friction µ , and crack half-length c.
During the course of this project, we have generated numerous
stress-strain plots (as
in Fig. 11 or 12) at different strain rates εr for various levels
and rates of prestrain. These
series provide values for the creep parameters in Equation 10, from
which we can predict
εD/B and compare with our experimentally observed transition strain
rates (Fig. 13).
The observed and predicted values for transition strain rate are
compared in Figure 15,
updated from Progress Report 2 with recalculated predictions for
εD/B. These updated
values are based on creep parameter derivations that have been
improved in two ways:
1) having additional data from tests conducted at more strain rates
over the past year, and
2) discarding a few ambiguous test results (e.g., low strain rate
tests that did not reach
a clear peak stress) that had been included previously. Another new
aspect of Figure 15
is that it now shows uncertainty as an error bar descending
from—instead of centered
on—each data point (colored symbol), which marks the lowest strain
rate at which brittle
behavior was observed. Thus the lower bound at the other end of
each error bar indicates
the highest strain rate at which ductile behavior was observed.
Given this uncertainty,
the experimental data for freshwater ice fit fairly closely to the
model, as do many of the
saline ice data. In one case for saline ice, the model
over-predicts εD/B by a factor of 30,
but this prediction was made from data of relatively few tests.
More tests at a range of εr
in the ductile regime are needed.
21
. '
1- 10 4•
predicted transition strain rate (s-1 )
Figure 15: Observed versus predicted values for to;s· Freshwater
data indicated in blue, saline data in green. Error bars indicate
the uncertainty in observed transition rate due to thus far varying
tr in half-decade increments.
4 Brittle Compressive Strength A set of tests has been run to study
the compressive strength of ice in the brittle regime, comparing
ice prestrained to 0.100 with undamaged specimens. The peak
stresses
2recorded during unjaxial compression at 1 x 10- s- 1 of eight
specimens each of prestrained and undamaged freshwater ice appear
in Figure 16. The points are staggered along the horizontal axis
for clarity, with the mean peak stress indicated in red for each
group. The means are practically the same. Only one of the
undamaged specimens reached a peak stress greater than 3 MPa
compared to three among the prestrained specimens. The values for
prestrained ice have slightly greater variance (1.1 versus 0.8
MPa), although the range from minimum to maximum stress is about
the same for both cases. Similar tests will be performed on saline
ice as well.
Overall, these data for freshwater ice do not show prestrain to
cause any significant difference in brittle compressive strength.
However, we caution that peak stress measured during a constant
strain rate compression is only one indicator of strength. At the
high strain rates (e.g., 1 x 10- 2 s- 1) needed to bring about
brittle failure, even the strongest specimens undergo stresses
above 1MPa for less than a fraction of a second. In contrast,
recall the tests we described in Section 3.2.2, in whlch
prestrained specimens were incapable of supporting stress held at
1.1 MPa (below any of the peak stresses in Fig. 16) for more than a
few rrunutes. We observed (1) that cracks nucleated and/or grew in
a prestrained specimen being held under those constant stresses,
and (2) that due to
22
−1Figure 16: Peak stress measured during compression at 1×10−2 s at
−10 C of freshwater ice, −1either (open circles) undamaged or
(filled squares) prestrained to 10 % at 1 ×10−5 s .
the pre-existing damage, cracks could link up with one another and
did not need to
propagate as far as in undamaged ice in order to fracture the
specimen. Thus, even if
after 0.10 prestrain the presence of damage—i.e., cracks—has
negligible effect on brittle
compressive strength with respect to loads of short (on the order
of seconds) duration,
damage may nevertheless weaken the capacity of ice to sustain
longer loads.
These observations highlight the complexity of damage effects in
relation to
creep and fracture phenomena in ice or to ice-structure
engineering, for example. We
recommend further research to investigate the concept of brittle
compressive strength of
damaged ice.
5 Publication A subset of work from this project was presented on
March 20th at the 13th International
Conference on the Physics and Chemistry of Ice (PCI-2014), hosted
here at the Thayer
School of Engineering in Hanover, New Hampshire. The talk focused
on the elastic
properties of freshwater ice, measured in the x1 direction
(parallel to the prestrain
direction). In that analysis, damage was quantified only in terms
of a scalar crack density
parameter. Following the PCI conference, we extended our treatment
of damage effects
on elastic properties to saline ice as well as freshwater ice, and
to the x2 (perpendicular to
prestrain) direction as well as x1. Additionally, we have developed
a crack density tensor
analysis (see Section 3.3 above) to quantify damage. These
advancements, summarized
in this report, are also being included in a manuscript which we
are about to finish shortly
and submit to a technical journal.
A second manuscript is being drafted to cover the topic of the
ductile-to-brittle
transition, specifically how damage affects the transition strain
rate. The observations of
ductile or brittle behavior upon reloading the prestrained
specimens are compared with
the predictions by the creep-versus-fracture model (see Section
3.5.1 above). As required,
we will provide BSEE with final drafts of all manuscripts prior to
their publication.
6 Equipment Status We are pleased to report no major mechanical
problems relating to the servo-hydraulic
true multi-axial test system (MATS) on which we perform all the
compression tests for
both prestraining and reloading. However, in this past year, the
freezer room that houses
the MATS has needed occasional repairs to the chiller system due to
leaking or ruptured
refrigerant lines, requiring a shut down period of usually no more
than a week.
The only major interruption to the work this year occurred with the
Hardinge
horizontal milling machine used to prepare the prismatic ice
specimens. The mill
operates by power from an electric motor transferred to the arbor
via a countershaft. The
bearings on the countershaft reached the end of their life and
failed. This led to a few
weeks of downtime for disassembly, cleaning, and procurement and
installation of new
pillow block bearings onto the countershaft, shown in Figure
17.
Figure 17: Horizontal milling machine (left) and new bearings
installed on countershaft (right).
24
7 Next Steps: (i) refine parameters for creep-vs-fracture model
(Eq. 10).
(ii) test brittle compressive strength of prestrained saline
ice.
(iii) impart damage through biaxial loading.
Most of the remaining work in this investigation will focus on how
prestrain affects
the ductile-to-brittle transition rate. We are presently focused on
filling in the gaps in the
stress-strain sequences (such as Figs. 11 and 12) in order to
better determine the creep
parameters, B and n, that fit into the model of Equation 10.
One of our next steps is to incorporate into the model a term for
damage-reduced
Young’s modulus, E ∗ . Taking a step back in the derivation of εtc,
the transition strain rate
occurs when the creep zone (of radius rc) around a crack tip equals
the zone in which
elastic strain dominates. That is,
2/(n−1) K2 (n + 1)2EnBt
rc ≈ I (11) 2πE2 2n
with the stress intensity factor KI = KIc at the ductile-to-brittle
transition. Through partial
differentiation, Equation 10 was obtained by assuming E ∗ ≈ E0, the
undamaged Young’s
modulus (Schulson and Duval, 2009). On the other hand, if we remove
that assumption,
after some algebra we have simply
E0 εD/B = εtc (12)
E∗
within which the fracture toughness KIc is also a function of the
reduced modulus.
An additional set of future tests will induce damage under biaxial
loading with the
objective of extending the range of prestrain that we have been
able to study through
uniaxial loading. Biaxial loading could allow for higher levels of
damage, perhaps
amplifying the effects we have seen so far.
References Kachanov, M. (1980), ‘Continuum model of medium with
cracks’, Journal of the
engineering mechanics division 106(5), 1039–1051.
Kachanov, M. (1992), ‘Effective elastic properties of cracked
solids: Critical review of
some basic concepts’, Applied Mechanics Reviews 45(8),
304–335.
Langleben, M. and Pounder, E. (1963), Ice and Snow Processes,
Properties, and Application, M.I.T. Press, Cambridge, Mass.,
chapter 7 Elastic parameters of sea ice,
pp. 69–78.
25
Renshaw, C. E. and Schulson, E. M. (2001), ‘Universal behaviour in
compressive failure
of brittle materials’, Nature 412(6850), 897–900.
Schulson, E. M. and Duval, P. (2009), Creep and Fracture of Ice,
Cambridge University
Press.
Timco, G. W. and Weeks, W. F. (2010), ‘A review of the engineering
properties of sea
ice’, Cold Regions Science and Technology 60(2), 107–129.
26
The Effect of Deformation Damage on the Mechanical Behavior of Sea
Ice: Ductile-to-Brittle Transition, Elastic Modulus, and Brittle.
Compressive Strength
1 Introduction