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