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Sliding friction of sea ice blocks
Cold Regions Science and Technology 76–77 (2012) 8–16
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Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o l d r e g i o n s
Sliding friction of sea ice blocks Robert S. Pritchard a,⁎, G. Stuart Knoke b,1, Douglas C. “Skip” Echert c,2 a b c
IceCasting, Inc., 20 Wilson Court, San Rafael, CA 94901–1230, United States ICF International, Inc., 12011 NE First Street, Suite 210, Bellevue, WA 98005, United States PIXOTEC, LLC, 15917 S.E. Fairwood Blvd., Renton, WA 98058, United States
a r t i c l e
i n f o
Article history: Received 3 February 2011 Accepted 5 April 2011 Keywords: Sea ice Coulomb friction Rafting Slip–stick Model Field test
a b s t r a c t Sliding friction is known to be the largest energy sink during ridging events and it controls the force during rafting. During the 1994 Sea Ice Mechanics Initiative (SIMI), we conducted a set of sliding friction tests by pulling meter-size blocks of sea ice over the natural ice cover to determine the friction coefficients. The ice blocks were similar in size to those occurring in natural rafting and ridging events. Our simple tests were conducted by pulling the blocks using a boat trailer winch and cable. This system was definitely compliant, and elasticity of the cable was an important property that gave rise to stick–slip behavior comparable to that observed during rafting and ridging events. A simple model was introduced to analyze the data. It proved useful in understanding the stick–slip response, and exposed the proper way to interpret results. Static and dynamic friction coefficients are presented for 25 test cases for blocks of various sizes pulled over ice covered with no snow and a small amount of snow. The model shows how to interpret the friction coefficients in this compliant system correctly. Interestingly, static and dynamic friction coefficients do not occur at the times of maximum and minimum pulling forces, respectively. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The experiments presented here were chosen as one of a set of experiments to identify and understand the sources of underwater noise that is generated by ice deformation events. Our focus here is on the mechanical behavior of ice blocks sliding along the ice cover. According to modeling studies, sliding friction is the largest contributor to energy dissipation during rafting and ridging events (Hopkins and Hibler, 1989; Pritchard, 1981). The earliest ridging models (Parmerter and Coon, 1972; Rothrock, 1975) recognized that frictional losses occur as the ice sheet is pushed into the sail and keel of a building ridge, but they used parameter values that kept them from recognizing that friction losses are the largest by far than all other sinks. The importance of friction led to our desire to derive friction coefficients from field measurements using sea ice cut into blocks that were roughly the size of blocks found in ridges. Upon observing a sliding stick–slip process, one is tempted to assume that the maximum force determines the static friction coefficient and the minimum force determines the dynamic friction coefficient. While this is true for systems that are infinitely stiff, we
⁎ Corresponding author. Tel./fax: + 1 415 454 9899 (W/H). E-mail addresses: [email protected] (R.S. Pritchard), sknoke@icfi.com (G.S. Knoke), [email protected] (D.C.“S.” Echert). URL: http://www.icecasting.com (R.S. Pritchard). 1 Tel.: + 1 425 688 0141x24 (W); fax: + 1 425 688 0180. 2 Tel.: + 1 425 255 0789; fax: + 1 425 917 0104. 0165-232X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2011.04.001
learned from this simple set of experiments that it is not true if the force is applied by a compliant loading frame. A simple spring and slider model demonstrates this behavior clearly. The results of our tests explain the stick–slip behavior that was observed during our experiments, and furthermore, they explain the stick–slip process that was observed during rafting events in the field. These tests were serendipitous in that we had a small budget and therefore used a small and compliant loading system. We were lucky that the parameters gave results having time signatures very similar to rafting events observed visually (and acoustically) during the Sea Ice Mechanics Initiative (SIMI) Experiment. What began as a straightforward field experiment to determine the sliding friction ended as an exciting data set that improved our understanding of the stick–slip behavior observed during sea ice rafting events. We used a standard steel towing cable pulled by a boat trailer winch to pull blocks of ice over and under the ice cover. We chose ice blocks of roughly meter-size cut from the local ice cover to approximate the blocks found most commonly in ridges. We did not know a priori that the inexpensive hardware was relatively compliant; that this compliance is essential to reproducing stick–slip behavior; nor would the lack of stiffness mimic the elastic behavior of the thin ice that normally participates in rafting events. Thus, serendipity worked in our favor to produce data that helps explain rafting behavior and the effect of elasticity of the surrounding ice cover. We have developed and tested a method of measuring both the static and dynamic coefficients of friction. Understanding the stick– slip behavior is important for both mechanical and acoustic reasons.
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9
Mechanically, the rafting of ice and the ride-up that occurs in the ridging process are important energy sinks. Acoustically, the slip–stick ice motion generates a large portion of the under-ice noise in the Arctic Ocean. Seismologists have seen similar slip–stick phenomena in seismic earthquake data. 2. Background Colbeck (1993) prepared a thorough bibliography on sliding friction of snow and ice. So then, why did we conduct these experiments? There were two reasons. First, we wanted to test ice blocks having same size and properties of ice during the field portion of SIMI. Second, we were also interested in determining acoustic signatures of ice processes that are active during rafting and ridging, although these data are not presented here. We observed stick–slip behavior in our tests. There are many examples of stick–slip behavior in the literature, under many different test conditions and for many different materials. Frederking and Barker (2002) also observed similar stick–slip behavior during friction tests of ice on construction materials. Their test frame was also compliant, and could be modeled in a way analogous to ours. Sammonds, et al. (2005) conducted double-direct-shear friction on floating ice sheets to determine the effects of frictional sliding on shear faults. They found stick–slip behavior that begins in a nucleation zone and propagates away as a wave packet. They also note that Rist (1997) has analyzed data to show that ice follows a nonlinear friction law in which the shear force is proportional to the normal force to the power of 2/3. This difference has no effect on our analytical method, but it would change the friction coefficients. Since our goals were simple, we were able to design a simple and inexpensive test setup that allowed us to determine the sliding friction properties for little cost. It was a typical towing cable pulled toward a fixed post by a boat trailer winch. This system was compliant because the cable was extensible under the applied loads. According to our analysis, this compliance led to the stick–slip behavior. The results are similar to those observed during ice rafting and ridging events (see Sammonds et al., 1998 and Sanderson, 1988). There are numerous studies of the physics of sliding behavior that focus on the small-scale detail of the interactions between two rough surfaces. We are not trying to learn about the physical details of this failure, but we are interested in the global response as ice blocks slide over one another. 3. Experimental setup The field experiments were made during April of 1994 from the SIMI ice camp. The camp was located in the Beaufort Sea, approximately 300 km north of Alaska. The ice blocks were cut from the extant ice cover. They were roughly 1 m2 in plan area and about 0.25 m thick. These blocks are typical of the size observed in ridges, and are about as large as could be manhandled. Block weights were obtained from load cell measurements after lifting the blocks with a tripod. Block weights depended on their geometry, except that we occasionally piled a second block on top of the test block to mimic thicker ice. For the friction measurements, a nylon cargo strap was wrapped around the block and connected to a 4.8 mm (3/16 in) diameter steel cable, which was pulled by a 12 V boat trailer winch. Fig. 1 shows a typical override test. Part way through the testing, a more secure method of attaching the yoke to the ice block was used. Mountaineering ice screws were affixed on opposite sides of the block. A steel yoke was attached to the screws. Experiments by other investigators at the camp involved ice and ocean acoustic measurements. To reduce winch noise being transferred to the ice, the winch was mounted on a sheet of plywood which was placed on an automotive tire. Fig. 2 shows the winch configura-
Fig. 1. Ice block being pulled by boat trailer winch and cable.
tion. We varied block size and thickness, snow cover, speed, and cable length during the tests. For the override tests, blocks were pulled across the level ice cover. The force history was measured using a standard load cell inserted between the loading strap and winch cable. Forces were digitally recorded using a Campbell Scientific CR10 Measurement and Control System. For a few tests, we also measured noise source history, but those data are not discussed here. For raft tests, blocks were pulled up cantilevered ice beams to simulate natural rafting. After cutting the test block from the ice, two parallel cuts, typically 7 m long, were made in the ice sheet beginning at the floating test block. This created a cantilevered beam approximately 1 m wide and 7 m long. Just before testing, the end of the beam was pushed down and the leading edge of the test block was placed on top of the beam. Winching of the test block then was started. As the ice block was pulled onto the beam, the weight of the block depressed the beam. Typically after the test block was pulled for 2 to 4 m, the cantilever broke, ending the rafting portion of that test. The length of the beam was selected to have it break naturally, away from the attached end of the beam. A picture taken at the conclusion of a raft test is shown in Fig. 3. For this test, snow was removed from the beam prior to the test. As can be seen in Fig. 3, a layer of slush was deposited on the beam as the test block passed over it. The slush was created by the grinding away of the soft, relatively warm, ice at the bottom of the test blocks. This slush was soft and became refrozen only after the end of each raft test. We did not measure ice temperature and so are not able to evaluate freezing between the block and ice cover. 4. Stick–slip behavior model A simple model is derived for a block pulled over an ice sheet by an elastic cable. The free body diagrams are shown in Fig. 4. The ice block has mass M, and it is subject to the pulling force F and the Coulomb friction force. While the block is stationary, the pulling force F is less than the maximum static friction force Fs = μ s Mg
ð1Þ
10
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Fig. 2. The boat trailer winch was anchored by a wooden post in the ice and rested on an automotive inner tube. Here the cable is led through a pulley and back to the anchor to reduce pulling speed by half.
where μs is the static friction coefficient between the block and the ice sheet, M is mass of the block, and g is the gravitational constant. When the static limit is reached, the block moves and inertial acceleration is balanced by the difference between pulling force and the sliding friction force M
dv = F μ Mg dt
ð2Þ
where v = ẋ is block speed, x is block position, and μ is the sliding friction coefficient. Friction coefficients were assumed constant during each test. The force exerted by the cable as it winds onto the winch can be written as F = Aσ, where A is cross-section area and σ is stress in the cable. The deformation rate d is the rate of extension divided by the cable length d = (w − v)/L, where w is the rate at which cable is
wound on the winch spool that is in a fixed position, v = ẋ is block speed, x is block position, and length L = Lo − x, where Lo is the initial length of the cable when F is zero. If the effective elasticity of the cable is E (as opposed to the elasticity of the cable material), the stress rate is related to the deformation rate by σ ̇ = Ed. Combining these relationships gives the rate of change of the cable pulling force as a function of the ice block motion and winch speed as F ̇ = AE
w−v : L
ð3Þ
We can combine the cable area, elastic constant, and length into one stiffness coefficient k = AE/L. The force deformation relationship is nonlinear because L = L(x) and both x and v are unknowns of the system of equations. While solving the nonlinear equations might be desirable, we elected to begin with a simple linear spring model of the cable. We recognize that the simple spring model neglects the facts that the cable can sag, oscillate up and down, and lay on the ice surface when tension is relieved, but its results are sufficient to interpret the field test data. Therefore, we simply linearize the force–deformation relationship by assuming that the cable length remains relatively constant during each slip–stick event. This assumption implies that L ≈ Lo − xo for some xo during each event. When the block is stationary, v = 0, the rate of increase of force is F ̇ = k w:
Fig. 3. Ice block and cantilever beam at the conclusion of a raft test. The ice block was harvested just beyond far end of the ice beam and then was pulled over the beam. The cantilever fractured at a point approximately half way along its length.
ð4Þ
Fig. 4. Free body diagram of towed ice block.
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When the block is sliding, the equation governing the force is derived by differentiating Eq. (3) and substituting the acceleration v̇ using Eq. (2). The result is
The velocity history during block motion is derived easily by rearranging the derivative of the force–displacement relationship (Eq. (3)) and substituting Eq. (4)
F ̈ + ω F = kẇ + μ kg
v=w+
2
ð5Þ
pffiffiffiffiffiffiffiffiffiffiffi where ω = k = M is the natural frequency of the system as it slides. We neglect changes in the pulling speed so ẇ = 0. The solution to Eq. (5) is straightforward. We choose a constant Fp as a particular solution Fp = μ Mg:
ð6Þ
The general solution is then written as the particular solution plus a harmonic function F = Fp + a cos ωðt−to Þ
ð7Þ
where a and to are arbitrary constants. The two arbitrary constants may be determined by forcing the solution through the intersection of the general dynamic solution and the stationary solution, which is the state when the block begins to slide. This state is indicated by s in Fig. 5. Here the force (Eq. (7)) is Fs = μ s Mg = Fp + a cos ωðts −to Þ
ð8Þ
and the time rate (Eq. (3)) is Fṡ = kðw−vs Þ = −ωa sin ωðts −to Þ:
ð9Þ
But the speed is zero here. The radius a is found to be 2
2
a = ½ðμ s −μ ÞMg +
2 kw ω
ð10Þ
ωa sin ωðt to Þ: k
ð13Þ
Velocity is out of phase from the force, but it is also a simple harmonic function. The motion history during block motion is determined similarly, but from the undifferentiated equation. We find x = xp
a cos ωðt−to Þ k
ð14Þ
where xp = ∫ wdt - Fp/k. The reference position xp moves in time as the cable shortens. The force history is most easily studied in phase space F; F ̇ . There the path is seen to follow an ellipse when sliding. To see this path, form F ̇ by differentiating Eq. (7), and manipulate the two trigonometric functions into the form
F−Fp a
2 +
Ḟ ωa
2 = 1:
ð15Þ
The force follows the path shown in Fig. 5. When stationary, the solution moves rightward along the horizontal line (in general it is nearly horizontal, it is exactly horizontal only if w is constant). The horizontal line is located along ordinate value kw. When moving, the solution state moves clockwise along the ellipse. The ellipse is centered at (Fp, 0). The half-length of the F-axis is a. The half-length of the F-̇ axis is ωa. The size of this ellipse depends on the loading rate w as well as the block mass M, and the friction coefficients μ and μs. As time advances and the solution reaches state e, the solution has values Fe = Fp −ðμ s −μ ÞMg = Fp + a cos ωðte −to Þ
ð16Þ
and the time (relative to the arbitrary to) satisfies either cos ωðts −to Þ =
and
ðμ s −μ ÞMg a
ð11Þ
or sin ωðts −to Þ = −
Fṡ = kðw−ve Þ = −ωa sin ωðte −to Þ:
But we may replace the sine function with the cosine function expression in Eq. (16) to find
kw : ωa
ð12Þ
5
h i1=2 2 kðw−ve Þ = −ωa 1− cos ωðte −to Þ 1=2 ðμ −μ ÞMg 2 = −ωa 1− − s a
The positive sign is appropriate at state e because the Fṡ must be positive here. Therefore,
3
dF/dt (kN/s)
2
ve = 0
1
e kw .0
ð19Þ
s
0 .2
.4
Fp
.6
Force F (kN)
-2 -3 -4
ð18Þ
= kw
4
-1
ð17Þ
m
-5 Fig. 5. Force history in phase space F; F ̇ .
o
.8
and the block stops at this state. This fact makes the path consistent because the next cycle can begin while the block is at rest. The phase space plot (Fig. 5) allows us to estimate parameter values easily, and to make several interesting observations about the solution. 1. First, the mass must be specified independently. 2. Second, the static friction coefficient can be estimated from the intersection of the two solutions, shown as point s in Fig. 5. 3. Third, the dynamic friction coefficient can be determined, but it is from the center of the ellipse (Fp, 0), not from the minimum force. 4. Fourth, the static friction force is not the largest force observed. Inertia of the block adds to the static friction force to give a maximum force Fp + a.
R.S. Pritchard et al. / Cold Regions Science and Technology 76–77 (2012) 8–16
5. Fifth, the cable stiffness can be estimated from the length of the axes after other parameters are determined, but it needs not be known to estimate friction coefficients. 6. Finally, we note that the dynamic solution is not valid when the radius is so large that force becomes negative. The cable cannot support compression in the experimental configuration. Elasticity of the towing cable is an essential property of this system. It controls the frequency of the stick–slip events. Thus, the event-scale elasticity of the ice cover is an essential property that strongly affects the stick–slip behavior observed during rafting and ridging events. Eventscale here means the size of a ridging or rafting event, say hundreds of meters. A separate analysis would be required to determine the event scale, which likely depends upon water drag, ice thickness as a function of the distance from the rafting event, other breaks in the ice sheet, and perhaps the wave speed of compression–expansion waves moving away from the rafting event through the ice sheet. 5. Recurrence time The period of an individual stick–slip event is called the recurrence time following a similar seismological definition (Xie and Farmer, 1994). The force and motion solutions of the idealized stick–slip model allow a direct calculation of the recurrence time as a function of material parameters. The period of a stick–slip event is divided into two intervals: Tstatic, when the block sticks, and Tdynam, when the block slides. The two time intervals are calculated separately. When the block is stuck, the force follows the horizontal line rightward from state e to state s (Fig. 5). The force increases from Fe to Fs at the rate kw during the period Tstatic = ts - te. The static period is determined by integrating Eq. (4) ts
Fs Fe = ∫ k w dt:
ð20Þ
te
If the winch pull speed w is nearly constant during a stick–slip cycle, the integral becomes Fs Fe = kwTstatic :
ð21Þ
Substituting for the force difference gives Tstatic =
2ðμ s −μ ÞMg : kw
ð22Þ
3.313 .6
Tslide =
2 ½ωðto −ts Þ + ωðtm −to Þ ω
2 π −1 ðμ s −μ ÞMg cos + : = ω a 2 Alternately, the sliding period can be rewritten as Tslide =
2 π −1 kw sin + : ω a 2
3.625 4.688
1.125 2
1
3
4
5
Time (s) Fig. 6. Time history of tension in pulling cable during short period of two stick–slip events. Markers indicate data points at regular intervals of 16 per second.
Adding the period when the block is sliding to the period when it is static gives the stick–slip recurrence period T = Tstatic + Tslide. This expression depends on four factors if the winch speed is constant ẇ = 0 : the block weight Mg, pull rate w, spring stiffness k, and the difference between static and dynamics friction coefficients μs − μ. 6. Sample force history The pulling force history in the cable is shown in Fig. 6 for two stick–slip cycles during override test number DFF8-13. For this test, the force was measured sixteen times per second. Each data point is shown with a small square (▪) to display the rate of change of the force F. This sample begins at time t = 1.125 s and continues for 3.555 s until t = 4.688 s. Initially the force rises slowly and steadily as the cable is pulled taut. At about t = 3.313 s, the block slips and the cable tension quickly falls for three measurements. After slipping, the cable tension relaxes and the cable flops to the ice. The block becomes stationary and the cable force then rises until about t = 4.500 s. During this stationary period, the cable oscillates between flopping to the ice and snapping upward. Finally, the static friction force is reached, the block slides, and the force quickly falls. The cable force history can be understood better by looking at the force phase plot shown in Fig. 7. The darker line shows the same force history data presented in Fig. 6, but here time rate of change Fṅ = F n+1 −F n−1 = t n+1 −t n−1 is plotted against the force Fn for a selected set of data points indicated by n. The same data points 5 4 3 2
dF/dt (kN/s)
The period from m to e is equal to the period from s through o to m. Therefore, the sliding period is
.4
.0
ð23Þ
Fm = Fp = Fp + a cos ωðtm −to Þ:
4.625
.2
When the block slides, the force follows the elliptical path clockwise in phase space from point s clockwise around to point e (Fig. 5). Because the functional values are cyclical and change sign, we must exercise care so we calculate the sliding period piecewise, from s to o, from o to m, and from m to e. The forces at states s and m are Fs = μ s Mg = Fp + a cos ωðts −to Þ
4.500
.8
Force (kN)
12
3.625 3.313
1
4.500
1.125
0 .0
.2
-1
.4
.6
.8
Force F (kN)
-2
ð24Þ
-3
4.688
-4 -5
ð25Þ
4.625
Fig. 7. Phase plot of cable force. The rate of change of cable force is plotted against the force.
R.S. Pritchard et al. / Cold Regions Science and Technology 76–77 (2012) 8–16
3
Force (kN)
identified by circle markers were identified in Fig. 6. In Fig. 7, we can see clearly that initially the force rises slowly and steadily as the cable is pulled taut and there is a rather small oscillation of the cable tension as the cable flops up and down. Then, at t = 3.313 s, the block slips and the cable tension quickly falls for four measurements as seen by the clockwise circular path. The cable flopping is seen in the three loops beginning at one point prior to t = 3.625 s and continuing until t = 4.500 s. At this latter time, the static friction limit is again reached, the block slides, and the force quickly falls. Two other curves appear in Fig. 7 (the two curves appearing in Fig. 5). These describe the idealized model developed to analyze the field data. The static loading curve is the horizontal line through F ̇ = 0:36 kN = s. We have assumed it constant and neglected any loading rate changes due to winding of the cable onto the winch spool. The dynamic loading curve is the ellipse. We have chosen the static friction coefficient to be μs = 1.24, using the larger second sticking event. The center of the horizontal axis is chosen to be μ = 0.78 so that the minimum force occurs approximately at the point before t = 3.625 s. The vertical axis of the ellipse can be determined if the cable stiffness coefficient is known, but it has only minimal effect on estimating the friction coefficients. Therefore, we have simply chosen a cable stiffness that fits the data well for these test conditions. Two surprising results become obvious from this phase plot, and they occur because the cable is compliant. First, the maximum cable tension occurs slightly after the block begins to slide because the cable is pulled at a constant rate but the block has not yet accelerated up to the pulling speed. Second, the dynamic friction coefficient is obtained from the force at the center of the ellipse, not the minimum force. This difference is quite large. Estimating the friction coefficients from the maximum and minimum values of the force history in Fig. 6 alone would give erroneous estimates of the correct values.
13
2 Test No. 6
1
Test No. 7 Avg
0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
Extension (m) Fig. 8. Cable stiffness test. The cable was pulled against a fixed post with force and extension history measured.
a block having mass M, the force pressing the block against the ice sheet is Mgcos α and the Coulomb friction force is − μMgcos α. When moving, inertia is balanced by the difference between the net pulling force F − Mgsin α and the Coulomb friction force, so that M v̇ = F−Mg sin α−μMg cos α:
ð26Þ
While the block is stationary, the pulling force depends on the extension as shown in Eq. (1) with the applied normal force (apparent weight) reduced to Mgcos α. With the forces modified by the sloped ice cover, the pulling force of the moving block is described by F ̈ + ω F = ω Mgð sinα + μ cos αÞ: 2
2
ð27Þ
7. Results Here the particular solution is chosen as Thirty-three field tests were conducted, with results of 25 presented here. Three different test conditions were used. Using a winch, blocks of ice were either pulled across the upper surface of the extant ice, pulled upward out of the water to mimic rafting, or pulled underwater along the bottom of the ice cover. The friction coefficients were estimated using the model presented in Section 4. The under-ice tests are beyond the scope of this paper, because they require consideration of a water drag force. The under-ice tests might be particularly useful in developing a ridge-building model because much more ice is subducted into the ridge keel than is rafted into its sail. Although recurrence times are not discussed in detail here, they were found to be consistent with the model presented in Section 5. Recurrence times depend primarily on the winch pull rate and stiffness because the fraction of time when the block sticks is much longer than the fraction of time when it slides. The times when sticking and sliding are given earlier in Eqs. (22) and (25). These recurrence times were also similar to those observed during SIMI in collaborative acoustic experiments with Xie and Farmer (1994). Of course, both sets of recurrence times are dependent on the loading rates, but it is interesting that they compared well during this period. pffiffiffiffiffiffiffiffiffiffiffi The frequency ω = k = M can be inferred by estimating k from Eq. (9) when both F ̇ and w are known, or it can be determined by measuring the stiffness k independently. We have done both. A separate test was run by pulling on a rigid post and measuring the pulling force as a function of distance stretched. The force history for this test is presented in Fig. 8. The measured stiffness was k = 23 kN/m. Since the cable had length L = 15 m, these tests lead to the elastic constant of AE = 352 kN. To handle the rafting tests, the simple model presented in Section 4 is modified for a block pulled over an ice sheet tilted upward at angle α. For
Fp = Mgðsinα + μ cos αÞ:
ð28Þ
and the general solution remains similar to Eq. (7). From the particular solution Fp, we can estimate the effect of the slope α. The cosine term reduces the effective mass and the sine term shifts the ellipse center to the right and increases the effective dynamic friction coefficient. How big are the changes? If the ice ramp has a rise of 0.20 m over a run of 2 m, then tan α = 0.10 and the other trigonometric functions are cos α ≈ 0.995 and sin α ≈ 0.0995. This would result in the dynamic friction coefficient being 10% too small if we neglect the slope α. If the ice ramp has a rise of only 0.30 m over a run of 5 m, then tan α = 0.06 and the other trigonometric functions are cos α ≈ 0.998 and sin α ≈ 0.060. This would result in the dynamic friction coefficient being 6% too small if we neglect the slope α. The static friction coefficient would be essentially unchanged because the apparent weight is nearly unchanged. In our judgment, these corrections are within our data uncertainty, so they are neglected here. At our higher pull rates, the force path in phase space appears to be elliptical, with no flat top indicating a sticking block. This result suggested that there might be a pull rate beyond which the block moves continuously and not as stick–slip motion. To address this question, we consider the block speed history described by the derivative of Eq. (1) with F ̇ given by Eq. (3) so that kðv−wÞ = ωa sin ωðt−to Þ:
ð29Þ
The minimum value of the block speed occurs when the sine function equals negative unity. Thus, kðv min −wÞ = ωað−1Þ:
ð30Þ
14
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0.80
0.80 Rafting and Over-ride tests are intermingled
0.70
Dynamic Friction Coefficient
Dynamic Friction Coefficient
No snow cover
0.60 0.50 0.40 0.30 0.20 0.10 0.00 0.0
Over-ride Raft Over-ride trendline Raft trendline
0.5
1.0
0.70 0.60 0.50 0.40 0.30 0.20
No snow cover Snow Cover No snow cover trendline Snow cover trendline
0.10 0.00
1.5
2.0
2.5
3.0
0
3.5
5
10
Ice Block Weight (kN) Fig. 9. Dynamic friction coefficient plotted versus ice block weight comparing rafting and over-ride with no snow cover.
Substituting for a using Eq. (10) and solving the equation for vmin shows that the minimum block speed is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v min = w− ðμ s −μ Þ2 g 2 = ω + w2 :
ð31Þ
Since the argument of the square root is always greater than w2, the square root always exceeds w and the minimum speed is therefore always negative. Since the speed becomes positive beginning at state s and continues to be positive after that, the speed must cross zero before becoming negative. We already know that it becomes stationary at state e. Thus, we have shown that the block comes to rest prior to reaching this minimum speed and the negative minimum speed cannot be reached. After coming to rest, the block must remain stationary because the cable cannot withstand compression. This derivation proves instead that the block always behaves in a stick–slip manner and it cannot slide continuously, no matter how large the pull rate w or how stiff the cable. The fraction of the recurrence time, however, becomes smaller at higher pull speeds with a stiffer cable. Fig. 9 shows that the dynamic friction coefficient from the rafting and over-ride tests varies in the same way with ice block weight for tests with no snow cover. Fig. 10 shows similar results for the static friction coefficient. The over-ride data point at a weight of 3 kN may not be accurate, but this data might indicate that the static friction
35
40
Rafting and Over-ride tests are intermingled
1.20
0.80
Static Friction Coefficient
Static Friction Coefficient
30
1.40
No snow cover
0.70 0.60 0.50 0.40 0.30
0.00 0.0
25
coefficient is somewhat lower for the rafting tests than for the override tests. Based on these data, the test results from rafting and override tests are intermingled in the following. The winch pull rate had little effect on the measured friction coefficients. Fig. 11 compares the dependence of dynamic friction coefficient on winch speed for tests with snow cover and no snow cover. Fig. 12 compares the dependence of static friction coefficient on winch pull rate for tests with snow cover and no snow cover. The snow cover was about 10 mm in all cases. Based on the data discussed in the previous figures, the test data for rafting and over-ride are intermingled in these charts. From these test results, we conclude that winch pull rate (up to 40 mm/s) has little impact on the measured friction coefficients. The ice block weight had a measurable effect on the friction coefficients. The heavier blocks exhibited lower friction coefficients. Fig. 13 shows the trend of dynamic friction coefficient with ice block weight, with snow cover and no snow cover; and Fig. 14 shows a similar trend for static friction coefficient with ice block weight, with snow cover and no snow cover. Based on the data discussed in the previous figures, the test data for rafting and over-ride are intermingled in these charts. These results are expected, given the behavior of ice under pressure, with the interface at the underside of a glacier being the extreme example. As the normal stress on the interface between the ice block and the ice surface increases, the
0.90
0.10
20
Fig. 11. Dynamic friction coefficient plotted against winch pull rate comparing snow cover and no snow cover, with rafting and over-ride tests intermingled.
1.00
0.20
15
Winch Pull Rate (mm/s)
Over-ride Raft Over-ride trendline
0.5
1.0
1.00 0.80 0.60 0.40 No snow cover Snow Cover
0.20
No snow cover trendline Snow cover trendline
1.5
2.0
2.5
3.0
3.5
Ice Block Weight (kN) Fig. 10. Static friction coefficient plotted versus ice block weight comparing rafting and over-ride for no snow cover.
0.00 0
5
10
15
20
25
30
35
40
Winch Pull Rate (mm/s) Fig. 12. Static friction coefficient plotted versus winch pull rate comparing snow cover and no snow cover, with rafting and over-ride tests intermingled.
R.S. Pritchard et al. / Cold Regions Science and Technology 76–77 (2012) 8–16
0.80
Dynamic Friction Coefficient
Rafting and Over-ride tests are intermingled
0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0.0
No snow cover Snow cover No snow cover trendline Snow cover trendline
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Weight of Ice Block (kN) Fig. 13. Dynamic friction coefficient plotted versus ice block weight comparing snow cover and no snow cover, with rafting and over-ride tests intermingled.
natural roughness of the two surfaces is smoothed out, yielding lower friction coefficients. Similarly, the drag coefficient of the light snow cover is less for the larger block. This result appears consistent with Rist (1997). To provide guidance for future modeling, the average dynamic and static friction coefficients without snow cover are 0.51 and 0.76, and with snow cover are 0.62 and 1.0, with overall averages of 0.56 and 0.91, respectively.
8. Discussion and conclusions A set of ice block towing tests was conducted. Ice blocks were cut from the extant ice cover and then towed either over the ice or rafted up onto the ice. Ice blocks were of order 1-meter square and about 0.30 m thick. These dimensions are typical of many blocks found in ridges. A standard boat trailer winch and towing cable were used, with pull rates of order 20 mm/s. Again, this speed is similar to that occurring in natural rafting events. The winch pull force was measured with a load cell. The force history was modeled using an elastic spring pulling against a rigid block with Coulomb friction force acting as drag. This simple model described the observed behavior accurately. Solutions
1.40 Rafting and Over-ride tests are intermingled
Static Friction Coefficient
1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.0
No snow cover Snow cover No snow cover trendline Snow cover trendline
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Weight of Ice Block (kN) Fig. 14. Static friction coefficient versus ice block weight comparing snow cover and no snow cover, with rafting and over-ride tests intermingled.
15
to this simple model are readily available. In addition to deriving the force history and the block speed history, we found it most useful to plot the force data in phase space F; F ̇ . Here the static solution is a straight horizontal line and the dynamic solution is an ellipse. The center of the ellipse is located at the dynamic friction force μ Mg, and the intersection of the static and dynamic solutions near maximum pulling force defines the static friction force μsMg. At the outset, we had no idea that the compliance of the tow cable would be an important property of these tests. It was, however, crucial to understand how the cable elasticity interacts with the Coulomb friction forces to give the stick–slip behavior. It was essential for proper interpretation of the pulling force data. In addition, we observed that natural rafting events also progress as stick–slip motions that have similar recurrence times as these block-towing tests. These were observed acoustically. We cannot infer elastic properties of the extant ice because the closing rates were not measured, but our results suggest an intriguing possible experiment to estimate large-scale elastic properties of the ice cover. Our initial supposition was that higher pull rates would cause the blocks to slide continuously rather than in stick–slip motion. The model showed this to be untrue; it predicts that stick–slip behavior will occur under all conditions. At higher pull speeds, most of the recurrence time was occupied during slipping and only a small fraction during sticking. This fact made interpretation confusing, until the model explained it. The field tests, although simplistic, provided a variety of results. The first (rather bland) result was that the friction coefficients obtained from rafting tests and over-ride tests were very similar. That is, the slope of the underlying ice had no effect on the friction coefficients. The second result is that winch pull rate (up to 40 mm/s) had little effect on the measured friction coefficients, with or without snow cover. The overall average dynamic and static friction coefficients were measured to be 0.56 and 0.91, respectively. The third result was that ice block weight had a measurable effect on the friction coefficients. The heavier blocks (such as those weighing 3 kN) exhibited about a third lower friction coefficients than lighter blocks (such as those weighing 0.3 kN). The fourth result was that the friction coefficients were larger with a snow cover, and the effect is more pronounced with the lighter blocks. For the lighter blocks (those weighing around 0.75 kN), the measured drag coefficient with light snow cover (~ 10 mm) was about 25% greater than for no snow cover. However, the light snow cover had little effect on the friction coefficients measured for the heavier blocks (those weighing around 2.25 kN). Also, the stick–slip behavior was more consistent from one cycle to the next when snow was present. To provide guidance for future modeling, the average dynamic and static friction coefficients without snow cover are 0.51 and 0.76, and with snow cover are 0.62 and 1.0, respectively. The sliding friction model can be applied to large-scale rafting by ˜ (stress integrated relating the towing force F to the stress resultant σ through ice thickness h) used in large-scale ice dynamics models. We ˜ where z is width of the ice block. The weight of the block have F = σz, is Mg = ρghzl, where ρ is mass density, g is the gravity constant, and l is the overlap distance (the remaining horizontal dimension of the block). Relating these expressions by the Coulomb friction law ˜ = μ ρghl. During rafting, this (F = μ Mg) gives the stress resultant σ force can increase as overlap distance increases until the thin ice sheet buckles. Rothrock (1979) analyzed the forces associated with these different failure processes. The thickness h can represent either the thickness of extant thin ice participating in rafting or multiple thicknesses as the ice piles into ridges, thereby increasing the effective weight of the block. While this project began as a study to determine friction coefficients, the simple model led us to also focus on gaining a better
16
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understanding how the cable compliance affected behavior, and by inference, how ice sheet elasticity affects the behavior of natural rafting events. Acknowledgments This project had the overarching purpose of understanding how noise is generated by deforming sea ice. That energy dissipated by different mechanical processes might serve as a good proxy for noise source levels was an idea conceived by our good friend and colleague Max Coon. In the earliest stages of processing these stick–slip data, Max was trying to find a way to identify chaotic behavior in one or another ice failure process. Thus, he suggested plotting the force versus the force delayed by a time step. His suggestion eventually led to our phase plots of the time rate of force versus the force. This idea led to rewriting the model solutions in this phase space, and to the comprehensive analysis presented in this paper. Once again, Max had made a wise, early suggestion that provided very important guidance to our work. RSP gratefully acknowledges the Office of Naval Research for funding this work under contracts N00014-92-C-0052, N00014-96-C-0174, and N00014-98-C-0810. RSP thanks Drs. Thomas Curtin and Michael Van Woert for their support.
References Colbeck, S.C., 1993. Bibliography on snow and ice friction. CRREL Report No: SR 93-06, 1-17, U.S. Army Cold Regions Research and Engineering Laboratory, Lyme, NH. Frederking, R., Barker, A., 2002. Friction of sea ice on various construction materials. 16th International Symposium on Ice, Dunedin, NZ, 2–6 December, pp. 442–449. Hopkins, M.A., Hibler III, W.D., 1989. On modeling the energetics of the ridging process. 8th International Conference on Offshore Mechanics and Arctic Engineering. American Society of Mechanical Engineers, The Hague, pp. 175–178. Mar 19–23. Parmerter, R.R., Coon, M.D., 1972. A model of pressure ridge formation in sea ice. J. Geophys. Res. 77, 6565–6575. Pritchard, R.S., 1981. Mechanical behavior of pack ice. In: Selvadurai, A.P.S. (Ed.), Mechanics of Structured Media. Elsevier, Amsterdam, pp. 371–405. Rist, M.A., 1997. High-stress ice fracture and friction. J. Phys. Chem. B 101 (32), 6263–6266. Rothrock, D.A., 1975. The energetics of the plastic deformation of pack ice by ridging. J. Geophys. Res. 80 (33), 4514–4519. Rothrock, D.A., 1979. Modeling sea-ice features and processes. J. Glaciol. 24 (90), 359–375. Sammonds, P., Hatton, D., Feltham, D., Taylor, P., 2005. Experimental study of sliding friction and stick–slip on faults in floating ice sheets. 18th Int'l. Conf. on Port and Ocean Eng. Under Arctic Conditions, POAC '05, Clarkson Univ., Potsdam, NY, pp. 303–312. Sammonds, P.R., Murrell, S.A.F., Rist, M.A., 1998. Fracture of multiyear sea ice. J. Geophys. Res. 103, 21,795–21,815. Sanderson, T.J.O., 1988. Ice Mechanics and Risks to Offshore Structures. Kluwer Academic Publishers, Dordrecht. Xie, Y., Farmer, D.M., 1994. Acoustic and seismic perspective of ice events observed in SIMI '94 experiment. Report to Office of Naval Research. The Institute of Ocean Sciences, Sidney, BC. Dec.
Contents lists available at ScienceDirect
Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o l d r e g i o n s
Sliding friction of sea ice blocks Robert S. Pritchard a,⁎, G. Stuart Knoke b,1, Douglas C. “Skip” Echert c,2 a b c
IceCasting, Inc., 20 Wilson Court, San Rafael, CA 94901–1230, United States ICF International, Inc., 12011 NE First Street, Suite 210, Bellevue, WA 98005, United States PIXOTEC, LLC, 15917 S.E. Fairwood Blvd., Renton, WA 98058, United States
a r t i c l e
i n f o
Article history: Received 3 February 2011 Accepted 5 April 2011 Keywords: Sea ice Coulomb friction Rafting Slip–stick Model Field test
a b s t r a c t Sliding friction is known to be the largest energy sink during ridging events and it controls the force during rafting. During the 1994 Sea Ice Mechanics Initiative (SIMI), we conducted a set of sliding friction tests by pulling meter-size blocks of sea ice over the natural ice cover to determine the friction coefficients. The ice blocks were similar in size to those occurring in natural rafting and ridging events. Our simple tests were conducted by pulling the blocks using a boat trailer winch and cable. This system was definitely compliant, and elasticity of the cable was an important property that gave rise to stick–slip behavior comparable to that observed during rafting and ridging events. A simple model was introduced to analyze the data. It proved useful in understanding the stick–slip response, and exposed the proper way to interpret results. Static and dynamic friction coefficients are presented for 25 test cases for blocks of various sizes pulled over ice covered with no snow and a small amount of snow. The model shows how to interpret the friction coefficients in this compliant system correctly. Interestingly, static and dynamic friction coefficients do not occur at the times of maximum and minimum pulling forces, respectively. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The experiments presented here were chosen as one of a set of experiments to identify and understand the sources of underwater noise that is generated by ice deformation events. Our focus here is on the mechanical behavior of ice blocks sliding along the ice cover. According to modeling studies, sliding friction is the largest contributor to energy dissipation during rafting and ridging events (Hopkins and Hibler, 1989; Pritchard, 1981). The earliest ridging models (Parmerter and Coon, 1972; Rothrock, 1975) recognized that frictional losses occur as the ice sheet is pushed into the sail and keel of a building ridge, but they used parameter values that kept them from recognizing that friction losses are the largest by far than all other sinks. The importance of friction led to our desire to derive friction coefficients from field measurements using sea ice cut into blocks that were roughly the size of blocks found in ridges. Upon observing a sliding stick–slip process, one is tempted to assume that the maximum force determines the static friction coefficient and the minimum force determines the dynamic friction coefficient. While this is true for systems that are infinitely stiff, we
⁎ Corresponding author. Tel./fax: + 1 415 454 9899 (W/H). E-mail addresses: [email protected] (R.S. Pritchard), sknoke@icfi.com (G.S. Knoke), [email protected] (D.C.“S.” Echert). URL: http://www.icecasting.com (R.S. Pritchard). 1 Tel.: + 1 425 688 0141x24 (W); fax: + 1 425 688 0180. 2 Tel.: + 1 425 255 0789; fax: + 1 425 917 0104. 0165-232X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2011.04.001
learned from this simple set of experiments that it is not true if the force is applied by a compliant loading frame. A simple spring and slider model demonstrates this behavior clearly. The results of our tests explain the stick–slip behavior that was observed during our experiments, and furthermore, they explain the stick–slip process that was observed during rafting events in the field. These tests were serendipitous in that we had a small budget and therefore used a small and compliant loading system. We were lucky that the parameters gave results having time signatures very similar to rafting events observed visually (and acoustically) during the Sea Ice Mechanics Initiative (SIMI) Experiment. What began as a straightforward field experiment to determine the sliding friction ended as an exciting data set that improved our understanding of the stick–slip behavior observed during sea ice rafting events. We used a standard steel towing cable pulled by a boat trailer winch to pull blocks of ice over and under the ice cover. We chose ice blocks of roughly meter-size cut from the local ice cover to approximate the blocks found most commonly in ridges. We did not know a priori that the inexpensive hardware was relatively compliant; that this compliance is essential to reproducing stick–slip behavior; nor would the lack of stiffness mimic the elastic behavior of the thin ice that normally participates in rafting events. Thus, serendipity worked in our favor to produce data that helps explain rafting behavior and the effect of elasticity of the surrounding ice cover. We have developed and tested a method of measuring both the static and dynamic coefficients of friction. Understanding the stick– slip behavior is important for both mechanical and acoustic reasons.
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9
Mechanically, the rafting of ice and the ride-up that occurs in the ridging process are important energy sinks. Acoustically, the slip–stick ice motion generates a large portion of the under-ice noise in the Arctic Ocean. Seismologists have seen similar slip–stick phenomena in seismic earthquake data. 2. Background Colbeck (1993) prepared a thorough bibliography on sliding friction of snow and ice. So then, why did we conduct these experiments? There were two reasons. First, we wanted to test ice blocks having same size and properties of ice during the field portion of SIMI. Second, we were also interested in determining acoustic signatures of ice processes that are active during rafting and ridging, although these data are not presented here. We observed stick–slip behavior in our tests. There are many examples of stick–slip behavior in the literature, under many different test conditions and for many different materials. Frederking and Barker (2002) also observed similar stick–slip behavior during friction tests of ice on construction materials. Their test frame was also compliant, and could be modeled in a way analogous to ours. Sammonds, et al. (2005) conducted double-direct-shear friction on floating ice sheets to determine the effects of frictional sliding on shear faults. They found stick–slip behavior that begins in a nucleation zone and propagates away as a wave packet. They also note that Rist (1997) has analyzed data to show that ice follows a nonlinear friction law in which the shear force is proportional to the normal force to the power of 2/3. This difference has no effect on our analytical method, but it would change the friction coefficients. Since our goals were simple, we were able to design a simple and inexpensive test setup that allowed us to determine the sliding friction properties for little cost. It was a typical towing cable pulled toward a fixed post by a boat trailer winch. This system was compliant because the cable was extensible under the applied loads. According to our analysis, this compliance led to the stick–slip behavior. The results are similar to those observed during ice rafting and ridging events (see Sammonds et al., 1998 and Sanderson, 1988). There are numerous studies of the physics of sliding behavior that focus on the small-scale detail of the interactions between two rough surfaces. We are not trying to learn about the physical details of this failure, but we are interested in the global response as ice blocks slide over one another. 3. Experimental setup The field experiments were made during April of 1994 from the SIMI ice camp. The camp was located in the Beaufort Sea, approximately 300 km north of Alaska. The ice blocks were cut from the extant ice cover. They were roughly 1 m2 in plan area and about 0.25 m thick. These blocks are typical of the size observed in ridges, and are about as large as could be manhandled. Block weights were obtained from load cell measurements after lifting the blocks with a tripod. Block weights depended on their geometry, except that we occasionally piled a second block on top of the test block to mimic thicker ice. For the friction measurements, a nylon cargo strap was wrapped around the block and connected to a 4.8 mm (3/16 in) diameter steel cable, which was pulled by a 12 V boat trailer winch. Fig. 1 shows a typical override test. Part way through the testing, a more secure method of attaching the yoke to the ice block was used. Mountaineering ice screws were affixed on opposite sides of the block. A steel yoke was attached to the screws. Experiments by other investigators at the camp involved ice and ocean acoustic measurements. To reduce winch noise being transferred to the ice, the winch was mounted on a sheet of plywood which was placed on an automotive tire. Fig. 2 shows the winch configura-
Fig. 1. Ice block being pulled by boat trailer winch and cable.
tion. We varied block size and thickness, snow cover, speed, and cable length during the tests. For the override tests, blocks were pulled across the level ice cover. The force history was measured using a standard load cell inserted between the loading strap and winch cable. Forces were digitally recorded using a Campbell Scientific CR10 Measurement and Control System. For a few tests, we also measured noise source history, but those data are not discussed here. For raft tests, blocks were pulled up cantilevered ice beams to simulate natural rafting. After cutting the test block from the ice, two parallel cuts, typically 7 m long, were made in the ice sheet beginning at the floating test block. This created a cantilevered beam approximately 1 m wide and 7 m long. Just before testing, the end of the beam was pushed down and the leading edge of the test block was placed on top of the beam. Winching of the test block then was started. As the ice block was pulled onto the beam, the weight of the block depressed the beam. Typically after the test block was pulled for 2 to 4 m, the cantilever broke, ending the rafting portion of that test. The length of the beam was selected to have it break naturally, away from the attached end of the beam. A picture taken at the conclusion of a raft test is shown in Fig. 3. For this test, snow was removed from the beam prior to the test. As can be seen in Fig. 3, a layer of slush was deposited on the beam as the test block passed over it. The slush was created by the grinding away of the soft, relatively warm, ice at the bottom of the test blocks. This slush was soft and became refrozen only after the end of each raft test. We did not measure ice temperature and so are not able to evaluate freezing between the block and ice cover. 4. Stick–slip behavior model A simple model is derived for a block pulled over an ice sheet by an elastic cable. The free body diagrams are shown in Fig. 4. The ice block has mass M, and it is subject to the pulling force F and the Coulomb friction force. While the block is stationary, the pulling force F is less than the maximum static friction force Fs = μ s Mg
ð1Þ
10
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Fig. 2. The boat trailer winch was anchored by a wooden post in the ice and rested on an automotive inner tube. Here the cable is led through a pulley and back to the anchor to reduce pulling speed by half.
where μs is the static friction coefficient between the block and the ice sheet, M is mass of the block, and g is the gravitational constant. When the static limit is reached, the block moves and inertial acceleration is balanced by the difference between pulling force and the sliding friction force M
dv = F μ Mg dt
ð2Þ
where v = ẋ is block speed, x is block position, and μ is the sliding friction coefficient. Friction coefficients were assumed constant during each test. The force exerted by the cable as it winds onto the winch can be written as F = Aσ, where A is cross-section area and σ is stress in the cable. The deformation rate d is the rate of extension divided by the cable length d = (w − v)/L, where w is the rate at which cable is
wound on the winch spool that is in a fixed position, v = ẋ is block speed, x is block position, and length L = Lo − x, where Lo is the initial length of the cable when F is zero. If the effective elasticity of the cable is E (as opposed to the elasticity of the cable material), the stress rate is related to the deformation rate by σ ̇ = Ed. Combining these relationships gives the rate of change of the cable pulling force as a function of the ice block motion and winch speed as F ̇ = AE
w−v : L
ð3Þ
We can combine the cable area, elastic constant, and length into one stiffness coefficient k = AE/L. The force deformation relationship is nonlinear because L = L(x) and both x and v are unknowns of the system of equations. While solving the nonlinear equations might be desirable, we elected to begin with a simple linear spring model of the cable. We recognize that the simple spring model neglects the facts that the cable can sag, oscillate up and down, and lay on the ice surface when tension is relieved, but its results are sufficient to interpret the field test data. Therefore, we simply linearize the force–deformation relationship by assuming that the cable length remains relatively constant during each slip–stick event. This assumption implies that L ≈ Lo − xo for some xo during each event. When the block is stationary, v = 0, the rate of increase of force is F ̇ = k w:
Fig. 3. Ice block and cantilever beam at the conclusion of a raft test. The ice block was harvested just beyond far end of the ice beam and then was pulled over the beam. The cantilever fractured at a point approximately half way along its length.
ð4Þ
Fig. 4. Free body diagram of towed ice block.
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11
When the block is sliding, the equation governing the force is derived by differentiating Eq. (3) and substituting the acceleration v̇ using Eq. (2). The result is
The velocity history during block motion is derived easily by rearranging the derivative of the force–displacement relationship (Eq. (3)) and substituting Eq. (4)
F ̈ + ω F = kẇ + μ kg
v=w+
2
ð5Þ
pffiffiffiffiffiffiffiffiffiffiffi where ω = k = M is the natural frequency of the system as it slides. We neglect changes in the pulling speed so ẇ = 0. The solution to Eq. (5) is straightforward. We choose a constant Fp as a particular solution Fp = μ Mg:
ð6Þ
The general solution is then written as the particular solution plus a harmonic function F = Fp + a cos ωðt−to Þ
ð7Þ
where a and to are arbitrary constants. The two arbitrary constants may be determined by forcing the solution through the intersection of the general dynamic solution and the stationary solution, which is the state when the block begins to slide. This state is indicated by s in Fig. 5. Here the force (Eq. (7)) is Fs = μ s Mg = Fp + a cos ωðts −to Þ
ð8Þ
and the time rate (Eq. (3)) is Fṡ = kðw−vs Þ = −ωa sin ωðts −to Þ:
ð9Þ
But the speed is zero here. The radius a is found to be 2
2
a = ½ðμ s −μ ÞMg +
2 kw ω
ð10Þ
ωa sin ωðt to Þ: k
ð13Þ
Velocity is out of phase from the force, but it is also a simple harmonic function. The motion history during block motion is determined similarly, but from the undifferentiated equation. We find x = xp
a cos ωðt−to Þ k
ð14Þ
where xp = ∫ wdt - Fp/k. The reference position xp moves in time as the cable shortens. The force history is most easily studied in phase space F; F ̇ . There the path is seen to follow an ellipse when sliding. To see this path, form F ̇ by differentiating Eq. (7), and manipulate the two trigonometric functions into the form
F−Fp a
2 +
Ḟ ωa
2 = 1:
ð15Þ
The force follows the path shown in Fig. 5. When stationary, the solution moves rightward along the horizontal line (in general it is nearly horizontal, it is exactly horizontal only if w is constant). The horizontal line is located along ordinate value kw. When moving, the solution state moves clockwise along the ellipse. The ellipse is centered at (Fp, 0). The half-length of the F-axis is a. The half-length of the F-̇ axis is ωa. The size of this ellipse depends on the loading rate w as well as the block mass M, and the friction coefficients μ and μs. As time advances and the solution reaches state e, the solution has values Fe = Fp −ðμ s −μ ÞMg = Fp + a cos ωðte −to Þ
ð16Þ
and the time (relative to the arbitrary to) satisfies either cos ωðts −to Þ =
and
ðμ s −μ ÞMg a
ð11Þ
or sin ωðts −to Þ = −
Fṡ = kðw−ve Þ = −ωa sin ωðte −to Þ:
But we may replace the sine function with the cosine function expression in Eq. (16) to find
kw : ωa
ð12Þ
5
h i1=2 2 kðw−ve Þ = −ωa 1− cos ωðte −to Þ 1=2 ðμ −μ ÞMg 2 = −ωa 1− − s a
The positive sign is appropriate at state e because the Fṡ must be positive here. Therefore,
3
dF/dt (kN/s)
2
ve = 0
1
e kw .0
ð19Þ
s
0 .2
.4
Fp
.6
Force F (kN)
-2 -3 -4
ð18Þ
= kw
4
-1
ð17Þ
m
-5 Fig. 5. Force history in phase space F; F ̇ .
o
.8
and the block stops at this state. This fact makes the path consistent because the next cycle can begin while the block is at rest. The phase space plot (Fig. 5) allows us to estimate parameter values easily, and to make several interesting observations about the solution. 1. First, the mass must be specified independently. 2. Second, the static friction coefficient can be estimated from the intersection of the two solutions, shown as point s in Fig. 5. 3. Third, the dynamic friction coefficient can be determined, but it is from the center of the ellipse (Fp, 0), not from the minimum force. 4. Fourth, the static friction force is not the largest force observed. Inertia of the block adds to the static friction force to give a maximum force Fp + a.
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5. Fifth, the cable stiffness can be estimated from the length of the axes after other parameters are determined, but it needs not be known to estimate friction coefficients. 6. Finally, we note that the dynamic solution is not valid when the radius is so large that force becomes negative. The cable cannot support compression in the experimental configuration. Elasticity of the towing cable is an essential property of this system. It controls the frequency of the stick–slip events. Thus, the event-scale elasticity of the ice cover is an essential property that strongly affects the stick–slip behavior observed during rafting and ridging events. Eventscale here means the size of a ridging or rafting event, say hundreds of meters. A separate analysis would be required to determine the event scale, which likely depends upon water drag, ice thickness as a function of the distance from the rafting event, other breaks in the ice sheet, and perhaps the wave speed of compression–expansion waves moving away from the rafting event through the ice sheet. 5. Recurrence time The period of an individual stick–slip event is called the recurrence time following a similar seismological definition (Xie and Farmer, 1994). The force and motion solutions of the idealized stick–slip model allow a direct calculation of the recurrence time as a function of material parameters. The period of a stick–slip event is divided into two intervals: Tstatic, when the block sticks, and Tdynam, when the block slides. The two time intervals are calculated separately. When the block is stuck, the force follows the horizontal line rightward from state e to state s (Fig. 5). The force increases from Fe to Fs at the rate kw during the period Tstatic = ts - te. The static period is determined by integrating Eq. (4) ts
Fs Fe = ∫ k w dt:
ð20Þ
te
If the winch pull speed w is nearly constant during a stick–slip cycle, the integral becomes Fs Fe = kwTstatic :
ð21Þ
Substituting for the force difference gives Tstatic =
2ðμ s −μ ÞMg : kw
ð22Þ
3.313 .6
Tslide =
2 ½ωðto −ts Þ + ωðtm −to Þ ω
2 π −1 ðμ s −μ ÞMg cos + : = ω a 2 Alternately, the sliding period can be rewritten as Tslide =
2 π −1 kw sin + : ω a 2
3.625 4.688
1.125 2
1
3
4
5
Time (s) Fig. 6. Time history of tension in pulling cable during short period of two stick–slip events. Markers indicate data points at regular intervals of 16 per second.
Adding the period when the block is sliding to the period when it is static gives the stick–slip recurrence period T = Tstatic + Tslide. This expression depends on four factors if the winch speed is constant ẇ = 0 : the block weight Mg, pull rate w, spring stiffness k, and the difference between static and dynamics friction coefficients μs − μ. 6. Sample force history The pulling force history in the cable is shown in Fig. 6 for two stick–slip cycles during override test number DFF8-13. For this test, the force was measured sixteen times per second. Each data point is shown with a small square (▪) to display the rate of change of the force F. This sample begins at time t = 1.125 s and continues for 3.555 s until t = 4.688 s. Initially the force rises slowly and steadily as the cable is pulled taut. At about t = 3.313 s, the block slips and the cable tension quickly falls for three measurements. After slipping, the cable tension relaxes and the cable flops to the ice. The block becomes stationary and the cable force then rises until about t = 4.500 s. During this stationary period, the cable oscillates between flopping to the ice and snapping upward. Finally, the static friction force is reached, the block slides, and the force quickly falls. The cable force history can be understood better by looking at the force phase plot shown in Fig. 7. The darker line shows the same force history data presented in Fig. 6, but here time rate of change Fṅ = F n+1 −F n−1 = t n+1 −t n−1 is plotted against the force Fn for a selected set of data points indicated by n. The same data points 5 4 3 2
dF/dt (kN/s)
The period from m to e is equal to the period from s through o to m. Therefore, the sliding period is
.4
.0
ð23Þ
Fm = Fp = Fp + a cos ωðtm −to Þ:
4.625
.2
When the block slides, the force follows the elliptical path clockwise in phase space from point s clockwise around to point e (Fig. 5). Because the functional values are cyclical and change sign, we must exercise care so we calculate the sliding period piecewise, from s to o, from o to m, and from m to e. The forces at states s and m are Fs = μ s Mg = Fp + a cos ωðts −to Þ
4.500
.8
Force (kN)
12
3.625 3.313
1
4.500
1.125
0 .0
.2
-1
.4
.6
.8
Force F (kN)
-2
ð24Þ
-3
4.688
-4 -5
ð25Þ
4.625
Fig. 7. Phase plot of cable force. The rate of change of cable force is plotted against the force.
R.S. Pritchard et al. / Cold Regions Science and Technology 76–77 (2012) 8–16
3
Force (kN)
identified by circle markers were identified in Fig. 6. In Fig. 7, we can see clearly that initially the force rises slowly and steadily as the cable is pulled taut and there is a rather small oscillation of the cable tension as the cable flops up and down. Then, at t = 3.313 s, the block slips and the cable tension quickly falls for four measurements as seen by the clockwise circular path. The cable flopping is seen in the three loops beginning at one point prior to t = 3.625 s and continuing until t = 4.500 s. At this latter time, the static friction limit is again reached, the block slides, and the force quickly falls. Two other curves appear in Fig. 7 (the two curves appearing in Fig. 5). These describe the idealized model developed to analyze the field data. The static loading curve is the horizontal line through F ̇ = 0:36 kN = s. We have assumed it constant and neglected any loading rate changes due to winding of the cable onto the winch spool. The dynamic loading curve is the ellipse. We have chosen the static friction coefficient to be μs = 1.24, using the larger second sticking event. The center of the horizontal axis is chosen to be μ = 0.78 so that the minimum force occurs approximately at the point before t = 3.625 s. The vertical axis of the ellipse can be determined if the cable stiffness coefficient is known, but it has only minimal effect on estimating the friction coefficients. Therefore, we have simply chosen a cable stiffness that fits the data well for these test conditions. Two surprising results become obvious from this phase plot, and they occur because the cable is compliant. First, the maximum cable tension occurs slightly after the block begins to slide because the cable is pulled at a constant rate but the block has not yet accelerated up to the pulling speed. Second, the dynamic friction coefficient is obtained from the force at the center of the ellipse, not the minimum force. This difference is quite large. Estimating the friction coefficients from the maximum and minimum values of the force history in Fig. 6 alone would give erroneous estimates of the correct values.
13
2 Test No. 6
1
Test No. 7 Avg
0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
Extension (m) Fig. 8. Cable stiffness test. The cable was pulled against a fixed post with force and extension history measured.
a block having mass M, the force pressing the block against the ice sheet is Mgcos α and the Coulomb friction force is − μMgcos α. When moving, inertia is balanced by the difference between the net pulling force F − Mgsin α and the Coulomb friction force, so that M v̇ = F−Mg sin α−μMg cos α:
ð26Þ
While the block is stationary, the pulling force depends on the extension as shown in Eq. (1) with the applied normal force (apparent weight) reduced to Mgcos α. With the forces modified by the sloped ice cover, the pulling force of the moving block is described by F ̈ + ω F = ω Mgð sinα + μ cos αÞ: 2
2
ð27Þ
7. Results Here the particular solution is chosen as Thirty-three field tests were conducted, with results of 25 presented here. Three different test conditions were used. Using a winch, blocks of ice were either pulled across the upper surface of the extant ice, pulled upward out of the water to mimic rafting, or pulled underwater along the bottom of the ice cover. The friction coefficients were estimated using the model presented in Section 4. The under-ice tests are beyond the scope of this paper, because they require consideration of a water drag force. The under-ice tests might be particularly useful in developing a ridge-building model because much more ice is subducted into the ridge keel than is rafted into its sail. Although recurrence times are not discussed in detail here, they were found to be consistent with the model presented in Section 5. Recurrence times depend primarily on the winch pull rate and stiffness because the fraction of time when the block sticks is much longer than the fraction of time when it slides. The times when sticking and sliding are given earlier in Eqs. (22) and (25). These recurrence times were also similar to those observed during SIMI in collaborative acoustic experiments with Xie and Farmer (1994). Of course, both sets of recurrence times are dependent on the loading rates, but it is interesting that they compared well during this period. pffiffiffiffiffiffiffiffiffiffiffi The frequency ω = k = M can be inferred by estimating k from Eq. (9) when both F ̇ and w are known, or it can be determined by measuring the stiffness k independently. We have done both. A separate test was run by pulling on a rigid post and measuring the pulling force as a function of distance stretched. The force history for this test is presented in Fig. 8. The measured stiffness was k = 23 kN/m. Since the cable had length L = 15 m, these tests lead to the elastic constant of AE = 352 kN. To handle the rafting tests, the simple model presented in Section 4 is modified for a block pulled over an ice sheet tilted upward at angle α. For
Fp = Mgðsinα + μ cos αÞ:
ð28Þ
and the general solution remains similar to Eq. (7). From the particular solution Fp, we can estimate the effect of the slope α. The cosine term reduces the effective mass and the sine term shifts the ellipse center to the right and increases the effective dynamic friction coefficient. How big are the changes? If the ice ramp has a rise of 0.20 m over a run of 2 m, then tan α = 0.10 and the other trigonometric functions are cos α ≈ 0.995 and sin α ≈ 0.0995. This would result in the dynamic friction coefficient being 10% too small if we neglect the slope α. If the ice ramp has a rise of only 0.30 m over a run of 5 m, then tan α = 0.06 and the other trigonometric functions are cos α ≈ 0.998 and sin α ≈ 0.060. This would result in the dynamic friction coefficient being 6% too small if we neglect the slope α. The static friction coefficient would be essentially unchanged because the apparent weight is nearly unchanged. In our judgment, these corrections are within our data uncertainty, so they are neglected here. At our higher pull rates, the force path in phase space appears to be elliptical, with no flat top indicating a sticking block. This result suggested that there might be a pull rate beyond which the block moves continuously and not as stick–slip motion. To address this question, we consider the block speed history described by the derivative of Eq. (1) with F ̇ given by Eq. (3) so that kðv−wÞ = ωa sin ωðt−to Þ:
ð29Þ
The minimum value of the block speed occurs when the sine function equals negative unity. Thus, kðv min −wÞ = ωað−1Þ:
ð30Þ
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0.80
0.80 Rafting and Over-ride tests are intermingled
0.70
Dynamic Friction Coefficient
Dynamic Friction Coefficient
No snow cover
0.60 0.50 0.40 0.30 0.20 0.10 0.00 0.0
Over-ride Raft Over-ride trendline Raft trendline
0.5
1.0
0.70 0.60 0.50 0.40 0.30 0.20
No snow cover Snow Cover No snow cover trendline Snow cover trendline
0.10 0.00
1.5
2.0
2.5
3.0
0
3.5
5
10
Ice Block Weight (kN) Fig. 9. Dynamic friction coefficient plotted versus ice block weight comparing rafting and over-ride with no snow cover.
Substituting for a using Eq. (10) and solving the equation for vmin shows that the minimum block speed is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v min = w− ðμ s −μ Þ2 g 2 = ω + w2 :
ð31Þ
Since the argument of the square root is always greater than w2, the square root always exceeds w and the minimum speed is therefore always negative. Since the speed becomes positive beginning at state s and continues to be positive after that, the speed must cross zero before becoming negative. We already know that it becomes stationary at state e. Thus, we have shown that the block comes to rest prior to reaching this minimum speed and the negative minimum speed cannot be reached. After coming to rest, the block must remain stationary because the cable cannot withstand compression. This derivation proves instead that the block always behaves in a stick–slip manner and it cannot slide continuously, no matter how large the pull rate w or how stiff the cable. The fraction of the recurrence time, however, becomes smaller at higher pull speeds with a stiffer cable. Fig. 9 shows that the dynamic friction coefficient from the rafting and over-ride tests varies in the same way with ice block weight for tests with no snow cover. Fig. 10 shows similar results for the static friction coefficient. The over-ride data point at a weight of 3 kN may not be accurate, but this data might indicate that the static friction
35
40
Rafting and Over-ride tests are intermingled
1.20
0.80
Static Friction Coefficient
Static Friction Coefficient
30
1.40
No snow cover
0.70 0.60 0.50 0.40 0.30
0.00 0.0
25
coefficient is somewhat lower for the rafting tests than for the override tests. Based on these data, the test results from rafting and override tests are intermingled in the following. The winch pull rate had little effect on the measured friction coefficients. Fig. 11 compares the dependence of dynamic friction coefficient on winch speed for tests with snow cover and no snow cover. Fig. 12 compares the dependence of static friction coefficient on winch pull rate for tests with snow cover and no snow cover. The snow cover was about 10 mm in all cases. Based on the data discussed in the previous figures, the test data for rafting and over-ride are intermingled in these charts. From these test results, we conclude that winch pull rate (up to 40 mm/s) has little impact on the measured friction coefficients. The ice block weight had a measurable effect on the friction coefficients. The heavier blocks exhibited lower friction coefficients. Fig. 13 shows the trend of dynamic friction coefficient with ice block weight, with snow cover and no snow cover; and Fig. 14 shows a similar trend for static friction coefficient with ice block weight, with snow cover and no snow cover. Based on the data discussed in the previous figures, the test data for rafting and over-ride are intermingled in these charts. These results are expected, given the behavior of ice under pressure, with the interface at the underside of a glacier being the extreme example. As the normal stress on the interface between the ice block and the ice surface increases, the
0.90
0.10
20
Fig. 11. Dynamic friction coefficient plotted against winch pull rate comparing snow cover and no snow cover, with rafting and over-ride tests intermingled.
1.00
0.20
15
Winch Pull Rate (mm/s)
Over-ride Raft Over-ride trendline
0.5
1.0
1.00 0.80 0.60 0.40 No snow cover Snow Cover
0.20
No snow cover trendline Snow cover trendline
1.5
2.0
2.5
3.0
3.5
Ice Block Weight (kN) Fig. 10. Static friction coefficient plotted versus ice block weight comparing rafting and over-ride for no snow cover.
0.00 0
5
10
15
20
25
30
35
40
Winch Pull Rate (mm/s) Fig. 12. Static friction coefficient plotted versus winch pull rate comparing snow cover and no snow cover, with rafting and over-ride tests intermingled.
R.S. Pritchard et al. / Cold Regions Science and Technology 76–77 (2012) 8–16
0.80
Dynamic Friction Coefficient
Rafting and Over-ride tests are intermingled
0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0.0
No snow cover Snow cover No snow cover trendline Snow cover trendline
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Weight of Ice Block (kN) Fig. 13. Dynamic friction coefficient plotted versus ice block weight comparing snow cover and no snow cover, with rafting and over-ride tests intermingled.
natural roughness of the two surfaces is smoothed out, yielding lower friction coefficients. Similarly, the drag coefficient of the light snow cover is less for the larger block. This result appears consistent with Rist (1997). To provide guidance for future modeling, the average dynamic and static friction coefficients without snow cover are 0.51 and 0.76, and with snow cover are 0.62 and 1.0, with overall averages of 0.56 and 0.91, respectively.
8. Discussion and conclusions A set of ice block towing tests was conducted. Ice blocks were cut from the extant ice cover and then towed either over the ice or rafted up onto the ice. Ice blocks were of order 1-meter square and about 0.30 m thick. These dimensions are typical of many blocks found in ridges. A standard boat trailer winch and towing cable were used, with pull rates of order 20 mm/s. Again, this speed is similar to that occurring in natural rafting events. The winch pull force was measured with a load cell. The force history was modeled using an elastic spring pulling against a rigid block with Coulomb friction force acting as drag. This simple model described the observed behavior accurately. Solutions
1.40 Rafting and Over-ride tests are intermingled
Static Friction Coefficient
1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.0
No snow cover Snow cover No snow cover trendline Snow cover trendline
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Weight of Ice Block (kN) Fig. 14. Static friction coefficient versus ice block weight comparing snow cover and no snow cover, with rafting and over-ride tests intermingled.
15
to this simple model are readily available. In addition to deriving the force history and the block speed history, we found it most useful to plot the force data in phase space F; F ̇ . Here the static solution is a straight horizontal line and the dynamic solution is an ellipse. The center of the ellipse is located at the dynamic friction force μ Mg, and the intersection of the static and dynamic solutions near maximum pulling force defines the static friction force μsMg. At the outset, we had no idea that the compliance of the tow cable would be an important property of these tests. It was, however, crucial to understand how the cable elasticity interacts with the Coulomb friction forces to give the stick–slip behavior. It was essential for proper interpretation of the pulling force data. In addition, we observed that natural rafting events also progress as stick–slip motions that have similar recurrence times as these block-towing tests. These were observed acoustically. We cannot infer elastic properties of the extant ice because the closing rates were not measured, but our results suggest an intriguing possible experiment to estimate large-scale elastic properties of the ice cover. Our initial supposition was that higher pull rates would cause the blocks to slide continuously rather than in stick–slip motion. The model showed this to be untrue; it predicts that stick–slip behavior will occur under all conditions. At higher pull speeds, most of the recurrence time was occupied during slipping and only a small fraction during sticking. This fact made interpretation confusing, until the model explained it. The field tests, although simplistic, provided a variety of results. The first (rather bland) result was that the friction coefficients obtained from rafting tests and over-ride tests were very similar. That is, the slope of the underlying ice had no effect on the friction coefficients. The second result is that winch pull rate (up to 40 mm/s) had little effect on the measured friction coefficients, with or without snow cover. The overall average dynamic and static friction coefficients were measured to be 0.56 and 0.91, respectively. The third result was that ice block weight had a measurable effect on the friction coefficients. The heavier blocks (such as those weighing 3 kN) exhibited about a third lower friction coefficients than lighter blocks (such as those weighing 0.3 kN). The fourth result was that the friction coefficients were larger with a snow cover, and the effect is more pronounced with the lighter blocks. For the lighter blocks (those weighing around 0.75 kN), the measured drag coefficient with light snow cover (~ 10 mm) was about 25% greater than for no snow cover. However, the light snow cover had little effect on the friction coefficients measured for the heavier blocks (those weighing around 2.25 kN). Also, the stick–slip behavior was more consistent from one cycle to the next when snow was present. To provide guidance for future modeling, the average dynamic and static friction coefficients without snow cover are 0.51 and 0.76, and with snow cover are 0.62 and 1.0, respectively. The sliding friction model can be applied to large-scale rafting by ˜ (stress integrated relating the towing force F to the stress resultant σ through ice thickness h) used in large-scale ice dynamics models. We ˜ where z is width of the ice block. The weight of the block have F = σz, is Mg = ρghzl, where ρ is mass density, g is the gravity constant, and l is the overlap distance (the remaining horizontal dimension of the block). Relating these expressions by the Coulomb friction law ˜ = μ ρghl. During rafting, this (F = μ Mg) gives the stress resultant σ force can increase as overlap distance increases until the thin ice sheet buckles. Rothrock (1979) analyzed the forces associated with these different failure processes. The thickness h can represent either the thickness of extant thin ice participating in rafting or multiple thicknesses as the ice piles into ridges, thereby increasing the effective weight of the block. While this project began as a study to determine friction coefficients, the simple model led us to also focus on gaining a better
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understanding how the cable compliance affected behavior, and by inference, how ice sheet elasticity affects the behavior of natural rafting events. Acknowledgments This project had the overarching purpose of understanding how noise is generated by deforming sea ice. That energy dissipated by different mechanical processes might serve as a good proxy for noise source levels was an idea conceived by our good friend and colleague Max Coon. In the earliest stages of processing these stick–slip data, Max was trying to find a way to identify chaotic behavior in one or another ice failure process. Thus, he suggested plotting the force versus the force delayed by a time step. His suggestion eventually led to our phase plots of the time rate of force versus the force. This idea led to rewriting the model solutions in this phase space, and to the comprehensive analysis presented in this paper. Once again, Max had made a wise, early suggestion that provided very important guidance to our work. RSP gratefully acknowledges the Office of Naval Research for funding this work under contracts N00014-92-C-0052, N00014-96-C-0174, and N00014-98-C-0810. RSP thanks Drs. Thomas Curtin and Michael Van Woert for their support.
References Colbeck, S.C., 1993. Bibliography on snow and ice friction. CRREL Report No: SR 93-06, 1-17, U.S. Army Cold Regions Research and Engineering Laboratory, Lyme, NH. Frederking, R., Barker, A., 2002. Friction of sea ice on various construction materials. 16th International Symposium on Ice, Dunedin, NZ, 2–6 December, pp. 442–449. Hopkins, M.A., Hibler III, W.D., 1989. On modeling the energetics of the ridging process. 8th International Conference on Offshore Mechanics and Arctic Engineering. American Society of Mechanical Engineers, The Hague, pp. 175–178. Mar 19–23. Parmerter, R.R., Coon, M.D., 1972. A model of pressure ridge formation in sea ice. J. Geophys. Res. 77, 6565–6575. Pritchard, R.S., 1981. Mechanical behavior of pack ice. In: Selvadurai, A.P.S. (Ed.), Mechanics of Structured Media. Elsevier, Amsterdam, pp. 371–405. Rist, M.A., 1997. High-stress ice fracture and friction. J. Phys. Chem. B 101 (32), 6263–6266. Rothrock, D.A., 1975. The energetics of the plastic deformation of pack ice by ridging. J. Geophys. Res. 80 (33), 4514–4519. Rothrock, D.A., 1979. Modeling sea-ice features and processes. J. Glaciol. 24 (90), 359–375. Sammonds, P., Hatton, D., Feltham, D., Taylor, P., 2005. Experimental study of sliding friction and stick–slip on faults in floating ice sheets. 18th Int'l. Conf. on Port and Ocean Eng. Under Arctic Conditions, POAC '05, Clarkson Univ., Potsdam, NY, pp. 303–312. Sammonds, P.R., Murrell, S.A.F., Rist, M.A., 1998. Fracture of multiyear sea ice. J. Geophys. Res. 103, 21,795–21,815. Sanderson, T.J.O., 1988. Ice Mechanics and Risks to Offshore Structures. Kluwer Academic Publishers, Dordrecht. Xie, Y., Farmer, D.M., 1994. Acoustic and seismic perspective of ice events observed in SIMI '94 experiment. Report to Office of Naval Research. The Institute of Ocean Sciences, Sidney, BC. Dec.