- Email: [email protected]
An atmospheric ice empirical failure criterion
Cold Regions Science and Technology 146 (2018) 81–86
Contents lists available at ScienceDirect
Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions
An atmospheric ice empirical failure criterion a,⁎
a
a
H. Farid , A. Saeidi , M. Farzaneh , F. Erchiqui a b
T
b
Canada Research Chair on Atmospheric Icing Engineering of Power Networks (INGIVRE), University of Quebec in Chicoutimi, Qc, Canada1 Laboratoire de Bioplasturig et Nanotechnologie, Université du Québec en Abitibi-Témiscamingue, Rouyn-Noranda, Québec, Canada
A R T I C L E I N F O
A B S T R A C T
Keywords: Atmospheric ice Brittle failure Ice shedding Empirical failure criterion
Characterizing the compressive strength of atmospheric ice is very significant to understand the ice shedding phenomenon. For this purpose, several tests were carried out in order to study the behavior of atmospheric ice under compression and tension, and under different experimental conditions. Ice was accumulated in a closed loop wind tunnel in order to simulate the natural processes of atmospheric icing. Four temperatures were considered (−20, −15, −10 and −5 °C), the wind speed inside the tunnel was set to 20 m/s in order to obtain a mean volume droplet diameter of 40 μm, and a water liquid content of 2.5 g/ m3. An empirical failure criterion for the atmospheric ice was proposed based on the experimental observations, taking into account the porosity, strain rate and temperature.
1. Introduction Atmospheric icing of structures is a phenomenon that affects human activities on many levels, and causes damages of different nature (Farzaneh, 2008). This phenomenon is manifested by a deposition of water drops or snowflakes on a cold surface (Farzaneh, 2000; Farzaneh, 2009). Structures in northern environments are often exposed to atmospheric icing. Wet snow, freezing rain, freezing fog, or hoarfrost are some of meteorological events resulting from atmospheric icing. Although the dangers caused by ice accretion on structures are considerable, ice shedding is particularly important. Ice shedding, which corresponds to the weight reduction of accumulated ice on a surface, is the source of several structural instabilities on overhead power line networks (Farzaneh, 2009)–(Fuheng & Shixiong, 1988). Understanding this phenomenon requires a deep knowledge of the structural and rheological properties of atmospheric ice. Unlike other types of ice such as fresh water ice or sea ice (Michel, 1978), few works on the mechanical properties of atmospheric ice are reported in the literature (Eskandarian, 2005)–(Druez et al., 1986). These contributions concern the study of the rheological characteristics of this type of ice, rather than providing quantification to its failure limits. Therefore, a presentation of failure criteria seems to be indispensable to understand the ice shedding phenomenon. The presentation of ice failure criteria was generally based on approaches concerning brittle materials, such as the maximum normal stress criterion (Schulson, 2001), the Mohr Coulomb criterion presented
⁎
1
by Schulson (Schulson, 2001) or recently by Farid et al. (Farid et al., 2017), the criterion of the maximum strain suggested by Hamza (Hamza, 1984), the strain energy criterion proposed by Cole (Cole, 1988), or the fracture toughness criterion for columnar grained ice S2 presented by Dempsey et al. (Dempsey & Wei, 1989). The failure of atmospheric ice involves many parameters that should be taken into account, this is more suggestive to the development of an empirical failure criterion specific to atmospheric ice. The compressive and tensile strength of atmospheric ice are considered to be the most significant properties for ice engineering, especially for the understanding of ice shedding mechanisms. These properties are highly affected by many environmental and structural parameters such as strain rate, temperature, wind speed, porosity, liquid water content, etc. (Kermani et al., 2007). Consequently, as compressive and tensile strength give a global image of the resistance of ice to different loads, they must be taken into account when considering the failure criterion. The present contribution concerns the development of an empirical failure criterion for atmospheric ice, which takes into account porosity, strain rate, and temperature effects. A significant correlation between the actual strength values and those predicted by the model was observed. The proposed criterion was validated in the case of compressive and tensile loadings.
Corresponding author. E-mail address: [email protected] (H. Farid). www.cigele.ca.
https://doi.org/10.1016/j.coldregions.2017.11.013 Received 11 April 2016; Received in revised form 29 October 2017; Accepted 20 November 2017 Available online 22 November 2017 0165-232X/ © 2017 Elsevier B.V. All rights reserved.
Cold Regions Science and Technology 146 (2018) 81–86
H. Farid et al.
0.016
2. Experimental procedure
0.014
2.1. Atmospheric ice preparation Strain (%)
0.012
The technique adopted to prepare atmospheric ice has an aim to reproduce the natural atmospheric icing process, specific conditions to generate this process were created in the atmospheric icing research wind tunnel at CIGELE Laboratories (Eskandarian, 2005; Kermani, 2007). Although the process has been detailed in a previous work by the same authors (Farid et al., 2016), explanation of the experimental procedure and setup seems important for the sake of clarity. Three independent supply lines provided air and water to the nozzles. Distilled water was injected into a cold airstream through nozzles located at the trailing edge of a spray bar. Air speed, water and air flux, all were controlled to generate droplets with a mean volume droplet diameter (MVD) of 40 μm and a liquid water content (LWC) of 2.5 g/ m3. A computer program allowed the control of these different parameters. Atmospheric ice was accumulated on a rotating aluminum cylinder (78-mm diameter and 590-mm length) making 1 rpm making the thickness distribution of ice uniform. The cylinder was carefully cleaned with hot water and soap before each set of experiment. Then it was placed at the middle of the test section of the wind tunnel. The distance between the cylinder and spray nozzles was large enough for the droplets to reach kinetic and thermodynamic equilibria. The time needed to grow a sufficient thickness of ice on the cylinder varied depending on accumulation conditions such as air temperature, velocity and liquid water content, it ranges from 2 to 4 h, sometimes up to 8 h. The atmospheric ice specimens were prepared at four different temperatures: − 20, −15, −10, and − 5 °C. The specimen orientation according to the accumulated ice is illustrated in Fig. 1. Once a thickness of about 60 mm was obtained, prismatic blocks were cut using a warm aluminum blade in order to avoid any mechanical stress. Then, the blocks were machined into cylindrical shape with a diameter of 40 mm and a length of 100 mm. These specimen dimensions were adopted in order to avoid any influence of the grain size on the compressive behavior of the ice (Schwarz et al., 1981). Before each test, the ice specimen ends were cleaned and smoothed to assure their parallelism. The specimen were fixed against the stainless steel platen using a thin piece of paper, the use of paper relax the triaxial stress field in the end of the specimen, and allowed to the test system to transfer the axial load perfectly. Once prepared, cylindrical specimens have been tested on uniaxial compression at the same temperature at which they have been accumulated. As strain rate in natural ice shedding is not more than 10− 2 s− 1 (Kermani et al., 2007), hence, four strain rates were chosen for the experimental tests: 10− 4 s− 1, 10− 3 s− 1, 10− 2 s− 1 and 10− 1 s− 1. The considered strain rates were calculated by dividing the test system cross head speed by the sample's length. The test system is considered infinitely stiff, so the system's compliance was negligible during the compressive tests (Farid et al., 2016).
0.01 0.008 0.006 0.004 0.002 0 0
50
100
150
200
Time (s) Fig. 2. Strain-time plot for a strain rate of 10− 4 s− 1 at a temperature of − 5 °C.
Plotting the strain versus time validated the claim that the specimen strain rates were constant during tests, this is shown in Fig. 2 below. In order to assure the results reproducibility, each test was repeated a minimum of five time, more details about the presented results are available in (Farid et al., 2016).
2.2. Microstructure observation and porosity evaluation Compared to other types of ice, atmospheric ice shows relatively high porosity and lower densities depending on the type of the accretion regime. In the present study, a Micro CT (Computed Tomography) has been used to quantify, with better accuracy, the porosity in the prepared samples (Farid et al., 2016). Based on the binary images presented in Fig. 3, one can note the influence of the accumulation temperature on the porosity, as temperature decreases, pores become smaller and their distribution become more uniform. The obtained values of porosity were 0.288 ± 0.079% at − 5 °C, 1.062 ± 0.082% at − 15 °C and 2.253 ± 0.089% at − 20 °C.
3. Experimental results For each experimental condition (temperature and strain rate), the stress versus strain curves were plotted, and the compressive strength, which corresponds the maximum stress reached before failure, was recorded. Fig. 4 shows the evolution of compressive strength of atmospheric ice as a function of strain rate for three temperatures: −20, − 15 and − 5 °C. The compressive strength of ice increases until it reaches a maximum value, which is then followed by a decrease as the strain rate increases. This transition characterizes the ductile-brittle transition, after which, the brittle failure takes part as dominant mode of failure, and the ice fractures without apparent plastic deformation. Fig. 5 illustrates the evolution of the compressive strength as a function of temperature at different strain rates. The highest values of atmospheric ice compressive strength were obtained for a temperature of − 15 °C. After this temperature is reached, the compressive strength decreases, which is related to the presence of more pores and cavities at temperature lower than − 15 °C. The evolution of porosity versus temperature is showed in Fig. 6. Porosity increases as the accumulation temperature decreases. This can be mainly related to the accretion mode; as temperature decreases, supercooled droplets reach the accumulation surface and freeze immediately upon contact, air particles get trapped in the interstices during this process, which therefore increases ice porosity.
Ice specimen
Aluminum cylinder
Atmospheric ice
Fig. 1. Schematic illustration of accumulated atmospheric ice and the specimen cut.
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Cold Regions Science and Technology 146 (2018) 81–86
H. Farid et al.
-20°C
-5°C
-15°C
Compressive strength (MPa)
Fig. 3. Binary image of cross sections of atmospheric ice at − 20 °C (left), − 15 °C (center) and − 5 °C (right).
9
Data at T = -20 °C
8
Data at T = -15 °C
7
Data at T = -5 °C
4. Empirical failure criterion development 4.1. Uniaxial compression failure criterion
6
Ice shedding by mechanical breaking of atmospheric ice takes place when the material reaches its rheological limits. Finding a failure criterion associated to this type of ice will be very helpful for engineering problems related to atmospheric icing. Atmospheric ice behavior is affected by many environmental and structural parameters, such as porosity, strain rate, and temperature, etc. Therefore, a good failure criterion should take into account these parameters. Strain rate is considered to be the most influential parameter of atmospheric ice compressive strength (Kermani et al., 2007). This provides the starting point for the mathematical derivation of an empirical failure criterion that predicts the strength of ice near the brittle failure domain. The compressive strength of atmospheric ice versus strain rate, as illustrated in Fig. 4, can be represented by a distribution similar to the normal law, taking into account the influence of strain rate and porosity, we can derive the following expression:
5 4 3 2 1 0 1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
Strain rate (s-1)
10-4 s-1
9
-3
-1
10 s
8
10-2 s-1
7
10-1 s-1
6 5 4 3 2 1
Compressive stregnth (MPa)
Fig. 4. Evolution of compressive strength versus strain rate for different temperatures.
2
σc =
-20
-15
-10
-5
0
Temperature (oC) Fig. 5. Compressive strength evolution as a function of temperature.
2.5
1
Porosity (%)
2 1.5
0.5 0 -20
-15
-10
(1)
where ε ̇ is the strain rate, η is the porosity, f(η) is a linear function of the porosity η, α is a constant depending on temperature, its dependence on temperature being evaluated using a temperature shift function, and n is a strain rate sensitivity factor. The average relative error between the model and experimental results are satisfactory, as shown in Fig. 7. Nonlinear regression was used for fitting the Eq. 1 to the experimental results in each temperature. The results are presented in the Table 1. The values obtained of the parameter α (alpha) were always close to the creep activation energy of ice (ranging between 45 and 90 kJ/mol (Schulson, 1999)). Which promote the hypothesis suggesting that this parameter could be related to the activation energy. Since this hypothesis needs furthers investigations, it is seen to be more convenient to express the parameter α just as an empirical parameter that depends on temperature. Plotting the evolution of the three parameters α, f(η) and n as functions of temperature, allows us to define their values in other ranges of temperature, as illustrated in Fig. 8(a)–(c). By plotting the Eq. (1) for the three temperatures, one can see that the three curves are closely parallel to each other (Fig. 7), which enhance the hypothesis that a shift function could be used to elucidate the temperature effect on the failure criterion as used by other authors (Sinha, 1978).
0 -25
α ⎛ 1 log (ε )̇ − n ⎞ ⎞ exp ⎜− ⎜⎛ ⎟ 2⎝ f (η) ⎠ ⎟ f (η) 2π ⎠ ⎝
-5
Temperature (oC) Fig. 6. Evolution of porosity versus temperature.
83
Cold Regions Science and Technology 146 (2018) 81–86
H. Farid et al. Data at T = -20 °C
9
Fig. 7. The empirical failure criterion versus experimental data.
Data at T = -15 °C
Compressive strength (MPa)
8
Data at T = -5 °C
7
Model at -20°C
6
Model at -15°C Model at -5°C
5 4 3 2 1 0 1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
80 70 60 50 40 30 20 10 0
Table 1 Different parameters presented in the failure criterion as a function of temperature. f(η)
α
− 5 °C − 15 °C − 20 °C
−6.71 −5.87 −6.98
2.99 3.22 3.35
33.14 67.31 54.65
The introduction of these type of functions requires the determination of the activation energy of atmospheric ice in different temperatures, which needs to further detailed investigation and was out of the scoop of this study.
-25
-20
-15
-10
-5
0
Temperature (oC)
(a): Evolution of parameter α as a function of temperature 3.4 3.35 3.3 3.25 3.2 3.15 3.1 3.05 3 2.95
4.2. Validation with compression tests data The proposed failure criterion was developed base on the experimental observations in three different temperatures (i.e − 5, − 15, and − 20 °C). In order to assure the reproducibility of the experimental results, as well as to verify the prediction capacity of the failure criterion in other ranges of temperature, the failure criterion parameters f (η), α, and n have been identified using non-linear regression based on Fig. 8(a)–(c) at −10 °C. The obtained values of the f(η), α, and n parameters are presented in Table 2. A separate set of experimental results at − 10 °C was used to compare the predicted values of compressive strength of atmospheric ice using the failure criterion with test results. Fig. 9 illustrates the experimental data at −10 °C versus the predicted values by the failure criterion. Fig. 9 shows an acceptable fitting between the failure criterion and the experimental data, the average relative error being 14.52%.
-25
-20
-15
-10
-5
f(η)
n
0
Temperature (oC)
(b): Evolution of parameter
( ) as a function of temperature
Temperature (oC) -25
-20
-15
-10
-5
0 0 -1 -2
4.3. Sensitivity analysis
-3 -4
In order to investigate the effect of the parameters α, f(η) and n on the failure criterion, a sensitivity analysis has been performed by ranging each of these parameters between a value of +20% and − 20% from its identified value (values in Table 1), while the two other parameters were set constant. As presented in Fig. 10, at −5 and − 15 °C, the failure criterion is sensitive to the parameter n and α, while it is less sensitive to f(η). At a colder temperature of − 20 °C, the failure criterion is highly sensitive to the three parameters.
-5
n Parameter
Temperature
α Parameter
Strain rate (s -1)
-6 -7 -8
(C): Evolution of parameter n as a function of temperature Fig. 8. Evolution of the empirical failure criterion parameters with temperature.
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Cold Regions Science and Technology 146 (2018) 81–86
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Table 3 Summary of results from uniaxial tensile tests at the different conditions (Mohamed & Farzaneh, 2011).
n
f(η)
α
− 6.52
3.11
46.83
7
Data at -10°C
6
Model at -10°C
Strain rate (s− 1)
4.44E-05 0.000111 0.000444 0.001111 0.001667
5
T = − 5 °C
T = − 10 °C
T = − 15 °C
Tensile strength (MPa)
Tensile strength (MPa)
Tensile strength (MPa)
1.42 1.42 1.27 1.23 1.19
1.44 1.43 1.34 1.33 1.27
1.50 1.45 1.39 1.36 1.30
4 3
1.6
2
1.4 Tensilte strength (MPa)
1 0 1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
Strain rate (s-1) Fig. 9. Failure criterion versus experimental data at − 10 °C.
Data -5°C Data -15°C Data -10°C Model -5°C Model -10°C Model -15°C
0.6 0.4 0.2
1.00E-05
1.00E-04
1.00E-03
1.00E-02
Strain rate (s -1)
Fig. 11. Evolution of the failure criterion and comparison with the experimental data for tensile strength.
Table 4 Parameters presented in the failure criterion as a function of temperature for tensile tests data.
4.4. Tensile failure criterion Ice shows brittle failure at a relatively slower strain rate in tension than in compression (Mohamed & Farzaneh, 2011). The proposed failure criterion, presented in Eq. (1), has been verified in the case of a tensile stress state in the brittle regime (for a strain rate higher than 10− 4 s− 1) using experimental data from Mohamed and Farzaneh (Mohamed & Farzaneh, 2011). Mohamed and Farzaneh tested the tensile strength of atmospheric ice under different temperatures, strain rates, and wind speed. Some of these results, presented in Table 3, were used for developing the failure criterion and for validation. Fig. 11 presents a comparison between the failure criterion and the experimental data at − 5, − 10, and − 15 °C. The failure criterion reproduces the tensile strength with good accuracy. The values of the parameter α, n and f(η) for tension failure criterion are listed in Table 4. The failure criterion predicts the tensile strength of atmospheric ice with good accuracy, the average relative errors being 1.48%, 1.19% and 0.72% for temperatures of − 5, − 10 and − 15 °C respectively.
n
α 20% f(η) -20% 2 4 6 Compressive Stregnth (MPa)
Temperature
n
f(η)
α
−5 °C −10 °C −15 °C
− 314.66 − 303.33 − 309.91
76.79 94.86 91.93
731,708 41,366.9 70,932.8
4.5. Validation with tensile tests data Other sets of experimental data from Mohamed and Farzaneh (Mohamed & Farzaneh, 2011) have been used for validation purposes as shown in Fig. 12. 5. Conclusions The objective of this work was to propose an empirical failure criterion for atmospheric ice as a function of different parameters such as temperature, strain rate and porosity. This failure criterion provides a good prediction of the compressive strength of atmospheric ice, the average relative errors associated to the model being 7.32, 6.66 and
Sensitivity analysis -15 o C Failure criterion parameters
Sensitivity analysis at -5 o C Failure criterion parameters
1 0.8
0
Since the failure criterion has been calibrated for temperatures ranging from − 5 to − 20 °C, and for strain rates higher than 10− 4 s− 1, and since we were more concerned about characterizing the atmospheric ice under ice shedding conditions, these intervals of temperature and strain rate constitute the range of its validity.
0
1.2
n
α 20% f(η) -20% 0 5 10 Compressive Stregnth (MPa)
Sensitivity analysis -20 o C Failure criterion parameters
Compressive stregnth (MPa)
Table 2 The failure criterion parameters identified at − 10 °C.
n
α 20% f(η) -20% 0
Fig. 10. Sensitivity analysis of the failure criterion at − 5, − 15 and − 20 °C.
85
5 10 Compressive Stregnth (MPa)
Cold Regions Science and Technology 146 (2018) 81–86
1.8
1.8
1.6
1.6
Tensile strength (MPa)
Tensile strength (MPa)
H. Farid et al.
1.4 1.2 1 0.8 Model -5°C
0.6
Series2
0.4
Series3
0.2
Series4 3.00E-04
1.2 1 0.8 Model -10°C Series2 Series3 Series4
0.6 0.4 0.2 0
0 3.00E-05
1.4
3.00E-05
3.00E-03
3.00E-04
3.00E-03 -1
Strain rate(s )
Strain rate (s -1)
(a)
(b)
Tensile strength (MPa)
1.8 1.6 1.4 1.2 1
Model -15°C
0.8
Series2
0.6
Series3
0.4
Series4
0.2 0 3.00E-05
3.00E-04 Strain rate (s -1)
3.00E-03
(c) Fig. 12. Validation of the failure criterion: (a) − 5 °C, (b) − 10 °C, (c) − 15 °C.
17.72% for temperatures − 5, − 15 and − 20 °C respectively. The proposed failure criterion was validated in the case of tensile stress state, the obtained results showed a good correlation between experimental data and the values obtained using the failure criterion.
Farid, H., Farzaneh, M., Saeidi, A., Erchiqui, F., Jan. 2016. A contribution to the study of the compressive behavior of atmospheric ice. Cold Reg. Sci. Technol. 121, 60–65. Farid, H., Saeidi, A., Farzaneh, M., 2017. Prediction of failure in atmospheric ice under triaxial compressive stress. Cold Reg. Sci. Technol. 138, 46–56. Farzaneh, M., Nov. 2000. Ice accretions on high-voltage conductors and insulators and related phenomena. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 358 (1776), 2971–3005. Farzaneh, M., 2008. Atmospheric Icing of Power Networks. Springer Science & Business Media. Farzaneh, M., Chisholm, W.A., 2009. Insulators for Icing and Polluted Environments. Fuheng, H., Shixiong, F., 1988. Icing on overhead the transmission lines in cold mountainous district of Southwest China and its protection. In: IWAIS, pp. 354–357. Hamza, H., 1984. Critical strain energy as a failure and crack propagation criterion for ice. In: Proc. IAHR Int. Symp. on Ice Problems. Kermani, M., 2007. Ice Shedding from Cables and Conductors – A Cracking Model of Atmospheric Ice. University of Quebec at Chicoutimi. Kermani, M., Farzaneh, M., Gagnon, R., Sep. 2007. Compressive strength of atmospheric ice. Cold Reg. Sci. Technol. 49 (3), 195–205. Michel, B., 1978. Ice Mechanics. Presses de l'université Laval. Mohamed, A.M.A., Farzaneh, M., Sep. 2011. An experimental study on the tensile properties of atmospheric ice. Cold Reg. Sci. Technol. 68 (3), 91–98. Schulson, E.M., Feb. 1999. The structure and mechanical behavior of ice. JOM 51 (2), 21–27. Schulson, E.M., Dec. 2001. Brittle failure of ice. Eng. Fract. Mech. 68 (17–18), 1839–1887. Schwarz, J., Frederking, R., Gavrillo, V., Petrov, I.G., Hirayama, K.-I., Mellor, M., Tryde, P., Vaudrey, K.D., Jul. 1981. Standardized testing methods for measuring mechanical properties of ice. Cold Reg. Sci. Technol. 4 (3), 245–253. Sinha, N.K., Jan. 1978. Short-term rheology of polycrystalline ice. J. Glaciol. 21 (85), 457–474.
Acknowledgement The present work was carried out within the frame work of the Canada Research Chair on Atrmospheric Icing Engineering of Power Networks (INGIVRE). The authors would like to thank all the sponsors of the project (Hydro-Québec, Hydro One, Réseau Transport d'Électricité (RTE), Alcan Cable, K-Line Insulators, Tyco Electronics, Dual-ADE, and FUQAC) whose financial support made this research possible. The authors also thank Mr. Pierre Camirand and Xavier Bouchard for the experimental setup. References Cole, D., 1988. Strain energy failure criterion for S2 fresh water ice in flexure. In: Proc. IAHR Int. Symp. on Ice Problems. vol. 1. pp. 206–215. Dempsey, J.P., Wei, Y., 1989. Fracture Toughness of S2 Columnar Freshwater ice: Crack Length and Specimen Size Effects - Part II. Druez, J., Nguyen, D.D., Lavoie, Y., Oct. 1986. Mechanical properties of atmospheric ice. Cold Reg. Sci. Technol. 13 (1), 67–74. Eskandarian, M., 2005. Ice Shedding from Overhead Electrical Lines by Mechanical Breaking.
86
Contents lists available at ScienceDirect
Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions
An atmospheric ice empirical failure criterion a,⁎
a
a
H. Farid , A. Saeidi , M. Farzaneh , F. Erchiqui a b
T
b
Canada Research Chair on Atmospheric Icing Engineering of Power Networks (INGIVRE), University of Quebec in Chicoutimi, Qc, Canada1 Laboratoire de Bioplasturig et Nanotechnologie, Université du Québec en Abitibi-Témiscamingue, Rouyn-Noranda, Québec, Canada
A R T I C L E I N F O
A B S T R A C T
Keywords: Atmospheric ice Brittle failure Ice shedding Empirical failure criterion
Characterizing the compressive strength of atmospheric ice is very significant to understand the ice shedding phenomenon. For this purpose, several tests were carried out in order to study the behavior of atmospheric ice under compression and tension, and under different experimental conditions. Ice was accumulated in a closed loop wind tunnel in order to simulate the natural processes of atmospheric icing. Four temperatures were considered (−20, −15, −10 and −5 °C), the wind speed inside the tunnel was set to 20 m/s in order to obtain a mean volume droplet diameter of 40 μm, and a water liquid content of 2.5 g/ m3. An empirical failure criterion for the atmospheric ice was proposed based on the experimental observations, taking into account the porosity, strain rate and temperature.
1. Introduction Atmospheric icing of structures is a phenomenon that affects human activities on many levels, and causes damages of different nature (Farzaneh, 2008). This phenomenon is manifested by a deposition of water drops or snowflakes on a cold surface (Farzaneh, 2000; Farzaneh, 2009). Structures in northern environments are often exposed to atmospheric icing. Wet snow, freezing rain, freezing fog, or hoarfrost are some of meteorological events resulting from atmospheric icing. Although the dangers caused by ice accretion on structures are considerable, ice shedding is particularly important. Ice shedding, which corresponds to the weight reduction of accumulated ice on a surface, is the source of several structural instabilities on overhead power line networks (Farzaneh, 2009)–(Fuheng & Shixiong, 1988). Understanding this phenomenon requires a deep knowledge of the structural and rheological properties of atmospheric ice. Unlike other types of ice such as fresh water ice or sea ice (Michel, 1978), few works on the mechanical properties of atmospheric ice are reported in the literature (Eskandarian, 2005)–(Druez et al., 1986). These contributions concern the study of the rheological characteristics of this type of ice, rather than providing quantification to its failure limits. Therefore, a presentation of failure criteria seems to be indispensable to understand the ice shedding phenomenon. The presentation of ice failure criteria was generally based on approaches concerning brittle materials, such as the maximum normal stress criterion (Schulson, 2001), the Mohr Coulomb criterion presented
⁎
1
by Schulson (Schulson, 2001) or recently by Farid et al. (Farid et al., 2017), the criterion of the maximum strain suggested by Hamza (Hamza, 1984), the strain energy criterion proposed by Cole (Cole, 1988), or the fracture toughness criterion for columnar grained ice S2 presented by Dempsey et al. (Dempsey & Wei, 1989). The failure of atmospheric ice involves many parameters that should be taken into account, this is more suggestive to the development of an empirical failure criterion specific to atmospheric ice. The compressive and tensile strength of atmospheric ice are considered to be the most significant properties for ice engineering, especially for the understanding of ice shedding mechanisms. These properties are highly affected by many environmental and structural parameters such as strain rate, temperature, wind speed, porosity, liquid water content, etc. (Kermani et al., 2007). Consequently, as compressive and tensile strength give a global image of the resistance of ice to different loads, they must be taken into account when considering the failure criterion. The present contribution concerns the development of an empirical failure criterion for atmospheric ice, which takes into account porosity, strain rate, and temperature effects. A significant correlation between the actual strength values and those predicted by the model was observed. The proposed criterion was validated in the case of compressive and tensile loadings.
Corresponding author. E-mail address: [email protected] (H. Farid). www.cigele.ca.
https://doi.org/10.1016/j.coldregions.2017.11.013 Received 11 April 2016; Received in revised form 29 October 2017; Accepted 20 November 2017 Available online 22 November 2017 0165-232X/ © 2017 Elsevier B.V. All rights reserved.
Cold Regions Science and Technology 146 (2018) 81–86
H. Farid et al.
0.016
2. Experimental procedure
0.014
2.1. Atmospheric ice preparation Strain (%)
0.012
The technique adopted to prepare atmospheric ice has an aim to reproduce the natural atmospheric icing process, specific conditions to generate this process were created in the atmospheric icing research wind tunnel at CIGELE Laboratories (Eskandarian, 2005; Kermani, 2007). Although the process has been detailed in a previous work by the same authors (Farid et al., 2016), explanation of the experimental procedure and setup seems important for the sake of clarity. Three independent supply lines provided air and water to the nozzles. Distilled water was injected into a cold airstream through nozzles located at the trailing edge of a spray bar. Air speed, water and air flux, all were controlled to generate droplets with a mean volume droplet diameter (MVD) of 40 μm and a liquid water content (LWC) of 2.5 g/ m3. A computer program allowed the control of these different parameters. Atmospheric ice was accumulated on a rotating aluminum cylinder (78-mm diameter and 590-mm length) making 1 rpm making the thickness distribution of ice uniform. The cylinder was carefully cleaned with hot water and soap before each set of experiment. Then it was placed at the middle of the test section of the wind tunnel. The distance between the cylinder and spray nozzles was large enough for the droplets to reach kinetic and thermodynamic equilibria. The time needed to grow a sufficient thickness of ice on the cylinder varied depending on accumulation conditions such as air temperature, velocity and liquid water content, it ranges from 2 to 4 h, sometimes up to 8 h. The atmospheric ice specimens were prepared at four different temperatures: − 20, −15, −10, and − 5 °C. The specimen orientation according to the accumulated ice is illustrated in Fig. 1. Once a thickness of about 60 mm was obtained, prismatic blocks were cut using a warm aluminum blade in order to avoid any mechanical stress. Then, the blocks were machined into cylindrical shape with a diameter of 40 mm and a length of 100 mm. These specimen dimensions were adopted in order to avoid any influence of the grain size on the compressive behavior of the ice (Schwarz et al., 1981). Before each test, the ice specimen ends were cleaned and smoothed to assure their parallelism. The specimen were fixed against the stainless steel platen using a thin piece of paper, the use of paper relax the triaxial stress field in the end of the specimen, and allowed to the test system to transfer the axial load perfectly. Once prepared, cylindrical specimens have been tested on uniaxial compression at the same temperature at which they have been accumulated. As strain rate in natural ice shedding is not more than 10− 2 s− 1 (Kermani et al., 2007), hence, four strain rates were chosen for the experimental tests: 10− 4 s− 1, 10− 3 s− 1, 10− 2 s− 1 and 10− 1 s− 1. The considered strain rates were calculated by dividing the test system cross head speed by the sample's length. The test system is considered infinitely stiff, so the system's compliance was negligible during the compressive tests (Farid et al., 2016).
0.01 0.008 0.006 0.004 0.002 0 0
50
100
150
200
Time (s) Fig. 2. Strain-time plot for a strain rate of 10− 4 s− 1 at a temperature of − 5 °C.
Plotting the strain versus time validated the claim that the specimen strain rates were constant during tests, this is shown in Fig. 2 below. In order to assure the results reproducibility, each test was repeated a minimum of five time, more details about the presented results are available in (Farid et al., 2016).
2.2. Microstructure observation and porosity evaluation Compared to other types of ice, atmospheric ice shows relatively high porosity and lower densities depending on the type of the accretion regime. In the present study, a Micro CT (Computed Tomography) has been used to quantify, with better accuracy, the porosity in the prepared samples (Farid et al., 2016). Based on the binary images presented in Fig. 3, one can note the influence of the accumulation temperature on the porosity, as temperature decreases, pores become smaller and their distribution become more uniform. The obtained values of porosity were 0.288 ± 0.079% at − 5 °C, 1.062 ± 0.082% at − 15 °C and 2.253 ± 0.089% at − 20 °C.
3. Experimental results For each experimental condition (temperature and strain rate), the stress versus strain curves were plotted, and the compressive strength, which corresponds the maximum stress reached before failure, was recorded. Fig. 4 shows the evolution of compressive strength of atmospheric ice as a function of strain rate for three temperatures: −20, − 15 and − 5 °C. The compressive strength of ice increases until it reaches a maximum value, which is then followed by a decrease as the strain rate increases. This transition characterizes the ductile-brittle transition, after which, the brittle failure takes part as dominant mode of failure, and the ice fractures without apparent plastic deformation. Fig. 5 illustrates the evolution of the compressive strength as a function of temperature at different strain rates. The highest values of atmospheric ice compressive strength were obtained for a temperature of − 15 °C. After this temperature is reached, the compressive strength decreases, which is related to the presence of more pores and cavities at temperature lower than − 15 °C. The evolution of porosity versus temperature is showed in Fig. 6. Porosity increases as the accumulation temperature decreases. This can be mainly related to the accretion mode; as temperature decreases, supercooled droplets reach the accumulation surface and freeze immediately upon contact, air particles get trapped in the interstices during this process, which therefore increases ice porosity.
Ice specimen
Aluminum cylinder
Atmospheric ice
Fig. 1. Schematic illustration of accumulated atmospheric ice and the specimen cut.
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-20°C
-5°C
-15°C
Compressive strength (MPa)
Fig. 3. Binary image of cross sections of atmospheric ice at − 20 °C (left), − 15 °C (center) and − 5 °C (right).
9
Data at T = -20 °C
8
Data at T = -15 °C
7
Data at T = -5 °C
4. Empirical failure criterion development 4.1. Uniaxial compression failure criterion
6
Ice shedding by mechanical breaking of atmospheric ice takes place when the material reaches its rheological limits. Finding a failure criterion associated to this type of ice will be very helpful for engineering problems related to atmospheric icing. Atmospheric ice behavior is affected by many environmental and structural parameters, such as porosity, strain rate, and temperature, etc. Therefore, a good failure criterion should take into account these parameters. Strain rate is considered to be the most influential parameter of atmospheric ice compressive strength (Kermani et al., 2007). This provides the starting point for the mathematical derivation of an empirical failure criterion that predicts the strength of ice near the brittle failure domain. The compressive strength of atmospheric ice versus strain rate, as illustrated in Fig. 4, can be represented by a distribution similar to the normal law, taking into account the influence of strain rate and porosity, we can derive the following expression:
5 4 3 2 1 0 1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
Strain rate (s-1)
10-4 s-1
9
-3
-1
10 s
8
10-2 s-1
7
10-1 s-1
6 5 4 3 2 1
Compressive stregnth (MPa)
Fig. 4. Evolution of compressive strength versus strain rate for different temperatures.
2
σc =
-20
-15
-10
-5
0
Temperature (oC) Fig. 5. Compressive strength evolution as a function of temperature.
2.5
1
Porosity (%)
2 1.5
0.5 0 -20
-15
-10
(1)
where ε ̇ is the strain rate, η is the porosity, f(η) is a linear function of the porosity η, α is a constant depending on temperature, its dependence on temperature being evaluated using a temperature shift function, and n is a strain rate sensitivity factor. The average relative error between the model and experimental results are satisfactory, as shown in Fig. 7. Nonlinear regression was used for fitting the Eq. 1 to the experimental results in each temperature. The results are presented in the Table 1. The values obtained of the parameter α (alpha) were always close to the creep activation energy of ice (ranging between 45 and 90 kJ/mol (Schulson, 1999)). Which promote the hypothesis suggesting that this parameter could be related to the activation energy. Since this hypothesis needs furthers investigations, it is seen to be more convenient to express the parameter α just as an empirical parameter that depends on temperature. Plotting the evolution of the three parameters α, f(η) and n as functions of temperature, allows us to define their values in other ranges of temperature, as illustrated in Fig. 8(a)–(c). By plotting the Eq. (1) for the three temperatures, one can see that the three curves are closely parallel to each other (Fig. 7), which enhance the hypothesis that a shift function could be used to elucidate the temperature effect on the failure criterion as used by other authors (Sinha, 1978).
0 -25
α ⎛ 1 log (ε )̇ − n ⎞ ⎞ exp ⎜− ⎜⎛ ⎟ 2⎝ f (η) ⎠ ⎟ f (η) 2π ⎠ ⎝
-5
Temperature (oC) Fig. 6. Evolution of porosity versus temperature.
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9
Fig. 7. The empirical failure criterion versus experimental data.
Data at T = -15 °C
Compressive strength (MPa)
8
Data at T = -5 °C
7
Model at -20°C
6
Model at -15°C Model at -5°C
5 4 3 2 1 0 1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
80 70 60 50 40 30 20 10 0
Table 1 Different parameters presented in the failure criterion as a function of temperature. f(η)
α
− 5 °C − 15 °C − 20 °C
−6.71 −5.87 −6.98
2.99 3.22 3.35
33.14 67.31 54.65
The introduction of these type of functions requires the determination of the activation energy of atmospheric ice in different temperatures, which needs to further detailed investigation and was out of the scoop of this study.
-25
-20
-15
-10
-5
0
Temperature (oC)
(a): Evolution of parameter α as a function of temperature 3.4 3.35 3.3 3.25 3.2 3.15 3.1 3.05 3 2.95
4.2. Validation with compression tests data The proposed failure criterion was developed base on the experimental observations in three different temperatures (i.e − 5, − 15, and − 20 °C). In order to assure the reproducibility of the experimental results, as well as to verify the prediction capacity of the failure criterion in other ranges of temperature, the failure criterion parameters f (η), α, and n have been identified using non-linear regression based on Fig. 8(a)–(c) at −10 °C. The obtained values of the f(η), α, and n parameters are presented in Table 2. A separate set of experimental results at − 10 °C was used to compare the predicted values of compressive strength of atmospheric ice using the failure criterion with test results. Fig. 9 illustrates the experimental data at −10 °C versus the predicted values by the failure criterion. Fig. 9 shows an acceptable fitting between the failure criterion and the experimental data, the average relative error being 14.52%.
-25
-20
-15
-10
-5
f(η)
n
0
Temperature (oC)
(b): Evolution of parameter
( ) as a function of temperature
Temperature (oC) -25
-20
-15
-10
-5
0 0 -1 -2
4.3. Sensitivity analysis
-3 -4
In order to investigate the effect of the parameters α, f(η) and n on the failure criterion, a sensitivity analysis has been performed by ranging each of these parameters between a value of +20% and − 20% from its identified value (values in Table 1), while the two other parameters were set constant. As presented in Fig. 10, at −5 and − 15 °C, the failure criterion is sensitive to the parameter n and α, while it is less sensitive to f(η). At a colder temperature of − 20 °C, the failure criterion is highly sensitive to the three parameters.
-5
n Parameter
Temperature
α Parameter
Strain rate (s -1)
-6 -7 -8
(C): Evolution of parameter n as a function of temperature Fig. 8. Evolution of the empirical failure criterion parameters with temperature.
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Table 3 Summary of results from uniaxial tensile tests at the different conditions (Mohamed & Farzaneh, 2011).
n
f(η)
α
− 6.52
3.11
46.83
7
Data at -10°C
6
Model at -10°C
Strain rate (s− 1)
4.44E-05 0.000111 0.000444 0.001111 0.001667
5
T = − 5 °C
T = − 10 °C
T = − 15 °C
Tensile strength (MPa)
Tensile strength (MPa)
Tensile strength (MPa)
1.42 1.42 1.27 1.23 1.19
1.44 1.43 1.34 1.33 1.27
1.50 1.45 1.39 1.36 1.30
4 3
1.6
2
1.4 Tensilte strength (MPa)
1 0 1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
Strain rate (s-1) Fig. 9. Failure criterion versus experimental data at − 10 °C.
Data -5°C Data -15°C Data -10°C Model -5°C Model -10°C Model -15°C
0.6 0.4 0.2
1.00E-05
1.00E-04
1.00E-03
1.00E-02
Strain rate (s -1)
Fig. 11. Evolution of the failure criterion and comparison with the experimental data for tensile strength.
Table 4 Parameters presented in the failure criterion as a function of temperature for tensile tests data.
4.4. Tensile failure criterion Ice shows brittle failure at a relatively slower strain rate in tension than in compression (Mohamed & Farzaneh, 2011). The proposed failure criterion, presented in Eq. (1), has been verified in the case of a tensile stress state in the brittle regime (for a strain rate higher than 10− 4 s− 1) using experimental data from Mohamed and Farzaneh (Mohamed & Farzaneh, 2011). Mohamed and Farzaneh tested the tensile strength of atmospheric ice under different temperatures, strain rates, and wind speed. Some of these results, presented in Table 3, were used for developing the failure criterion and for validation. Fig. 11 presents a comparison between the failure criterion and the experimental data at − 5, − 10, and − 15 °C. The failure criterion reproduces the tensile strength with good accuracy. The values of the parameter α, n and f(η) for tension failure criterion are listed in Table 4. The failure criterion predicts the tensile strength of atmospheric ice with good accuracy, the average relative errors being 1.48%, 1.19% and 0.72% for temperatures of − 5, − 10 and − 15 °C respectively.
n
α 20% f(η) -20% 2 4 6 Compressive Stregnth (MPa)
Temperature
n
f(η)
α
−5 °C −10 °C −15 °C
− 314.66 − 303.33 − 309.91
76.79 94.86 91.93
731,708 41,366.9 70,932.8
4.5. Validation with tensile tests data Other sets of experimental data from Mohamed and Farzaneh (Mohamed & Farzaneh, 2011) have been used for validation purposes as shown in Fig. 12. 5. Conclusions The objective of this work was to propose an empirical failure criterion for atmospheric ice as a function of different parameters such as temperature, strain rate and porosity. This failure criterion provides a good prediction of the compressive strength of atmospheric ice, the average relative errors associated to the model being 7.32, 6.66 and
Sensitivity analysis -15 o C Failure criterion parameters
Sensitivity analysis at -5 o C Failure criterion parameters
1 0.8
0
Since the failure criterion has been calibrated for temperatures ranging from − 5 to − 20 °C, and for strain rates higher than 10− 4 s− 1, and since we were more concerned about characterizing the atmospheric ice under ice shedding conditions, these intervals of temperature and strain rate constitute the range of its validity.
0
1.2
n
α 20% f(η) -20% 0 5 10 Compressive Stregnth (MPa)
Sensitivity analysis -20 o C Failure criterion parameters
Compressive stregnth (MPa)
Table 2 The failure criterion parameters identified at − 10 °C.
n
α 20% f(η) -20% 0
Fig. 10. Sensitivity analysis of the failure criterion at − 5, − 15 and − 20 °C.
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5 10 Compressive Stregnth (MPa)
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1.8
1.8
1.6
1.6
Tensile strength (MPa)
Tensile strength (MPa)
H. Farid et al.
1.4 1.2 1 0.8 Model -5°C
0.6
Series2
0.4
Series3
0.2
Series4 3.00E-04
1.2 1 0.8 Model -10°C Series2 Series3 Series4
0.6 0.4 0.2 0
0 3.00E-05
1.4
3.00E-05
3.00E-03
3.00E-04
3.00E-03 -1
Strain rate(s )
Strain rate (s -1)
(a)
(b)
Tensile strength (MPa)
1.8 1.6 1.4 1.2 1
Model -15°C
0.8
Series2
0.6
Series3
0.4
Series4
0.2 0 3.00E-05
3.00E-04 Strain rate (s -1)
3.00E-03
(c) Fig. 12. Validation of the failure criterion: (a) − 5 °C, (b) − 10 °C, (c) − 15 °C.
17.72% for temperatures − 5, − 15 and − 20 °C respectively. The proposed failure criterion was validated in the case of tensile stress state, the obtained results showed a good correlation between experimental data and the values obtained using the failure criterion.
Farid, H., Farzaneh, M., Saeidi, A., Erchiqui, F., Jan. 2016. A contribution to the study of the compressive behavior of atmospheric ice. Cold Reg. Sci. Technol. 121, 60–65. Farid, H., Saeidi, A., Farzaneh, M., 2017. Prediction of failure in atmospheric ice under triaxial compressive stress. Cold Reg. Sci. Technol. 138, 46–56. Farzaneh, M., Nov. 2000. Ice accretions on high-voltage conductors and insulators and related phenomena. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 358 (1776), 2971–3005. Farzaneh, M., 2008. Atmospheric Icing of Power Networks. Springer Science & Business Media. Farzaneh, M., Chisholm, W.A., 2009. Insulators for Icing and Polluted Environments. Fuheng, H., Shixiong, F., 1988. Icing on overhead the transmission lines in cold mountainous district of Southwest China and its protection. In: IWAIS, pp. 354–357. Hamza, H., 1984. Critical strain energy as a failure and crack propagation criterion for ice. In: Proc. IAHR Int. Symp. on Ice Problems. Kermani, M., 2007. Ice Shedding from Cables and Conductors – A Cracking Model of Atmospheric Ice. University of Quebec at Chicoutimi. Kermani, M., Farzaneh, M., Gagnon, R., Sep. 2007. Compressive strength of atmospheric ice. Cold Reg. Sci. Technol. 49 (3), 195–205. Michel, B., 1978. Ice Mechanics. Presses de l'université Laval. Mohamed, A.M.A., Farzaneh, M., Sep. 2011. An experimental study on the tensile properties of atmospheric ice. Cold Reg. Sci. Technol. 68 (3), 91–98. Schulson, E.M., Feb. 1999. The structure and mechanical behavior of ice. JOM 51 (2), 21–27. Schulson, E.M., Dec. 2001. Brittle failure of ice. Eng. Fract. Mech. 68 (17–18), 1839–1887. Schwarz, J., Frederking, R., Gavrillo, V., Petrov, I.G., Hirayama, K.-I., Mellor, M., Tryde, P., Vaudrey, K.D., Jul. 1981. Standardized testing methods for measuring mechanical properties of ice. Cold Reg. Sci. Technol. 4 (3), 245–253. Sinha, N.K., Jan. 1978. Short-term rheology of polycrystalline ice. J. Glaciol. 21 (85), 457–474.
Acknowledgement The present work was carried out within the frame work of the Canada Research Chair on Atrmospheric Icing Engineering of Power Networks (INGIVRE). The authors would like to thank all the sponsors of the project (Hydro-Québec, Hydro One, Réseau Transport d'Électricité (RTE), Alcan Cable, K-Line Insulators, Tyco Electronics, Dual-ADE, and FUQAC) whose financial support made this research possible. The authors also thank Mr. Pierre Camirand and Xavier Bouchard for the experimental setup. References Cole, D., 1988. Strain energy failure criterion for S2 fresh water ice in flexure. In: Proc. IAHR Int. Symp. on Ice Problems. vol. 1. pp. 206–215. Dempsey, J.P., Wei, Y., 1989. Fracture Toughness of S2 Columnar Freshwater ice: Crack Length and Specimen Size Effects - Part II. Druez, J., Nguyen, D.D., Lavoie, Y., Oct. 1986. Mechanical properties of atmospheric ice. Cold Reg. Sci. Technol. 13 (1), 67–74. Eskandarian, M., 2005. Ice Shedding from Overhead Electrical Lines by Mechanical Breaking.
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