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Multi-surface failure criterion for saline ice in the brittle regime
Cold Regions Science and Technology 36 (2003) 47 – 70 www.elsevier.com/locate/coldregions
Multi-surface failure criterion for saline ice in the brittle regime Ahmed Derradji-Aouat * Institute for Marine Dynamics, Memorial University Campus, National Research Council Canada, P.O. Box 12093, Station A, St. John’s, Newfoundland, Canada A1B 3T5 Received 24 April 2002; accepted 7 November 2002
Abstract In this paper, the development of a 3-D failure criterion for saline ice is presented. The need for such general 3-D failure formulation stems from the fact that, during ice – ship interactions, ice undergoes a complex state of deformation and stress before it fails and breaks away, and the use of the uniaxial strength of ice to compute impact ice loads may lead to inaccurate load calculations and non-conclusive results. In recent years, with the availability of High Power Computers (HPC), numerical methods are being used more than ever before in marine and ice engineering problems. Numerical models based on computational techniques such as finite elements, boundary elements and discrete elements require 3-D constitutive models and failure criteria to represent the behavior of the materials involved (such as the behavior of the ship structure, ice and water ‘‘fluid’’). At high-speed impacts (strain rates >10 3 s 1), ice behaves as a linear elastic material with a brittle mode of failure. Previously, Derradji-Aouat [Derradji-Aouat, A., 2000. A unified failure envelope for isotropic freshwater ice and iceberg ice. ASME/OMAE-2000, Int. Conference on Offshore Mechanics and Arctic Engineering, Polar and Arctic section, New Orleans, US, PDF file # OMAE-2000-P/A # 1002] developed a unified 3-D failure envelope for both fresh water isotropic ice and iceberg ice. In this paper, that formulation is extended to include failure of saline ice (in addition to fresh water ice and iceberg ice). The results of a significant number of true triaxial tests using Laboratory Grown Ice (LGSI) were obtained from the open literature. The results of these tests formed a database that enables the existing failure model [Derradji-Aouat, A., 2000. A unified failure envelope for isotropic freshwater ice and iceberg ice. ASME/OMAE-2000, Int. Conference on Offshore Mechanics and Arctic Engineering, Polar and Arctic section, New Orleans, US, PDF file # OMAE-2000-P/A # 1002] to be extended from the isotropic fresh water ice and iceberg ice to columnar saline ice. Mroz’s [J. Mech. Phys. Solids 15 (1967) 163] concept for the multi-surface failure theory is used in both studies (the present study, for saline ice, as well as in the previous study, for the fresh water isotropic ice and iceberg ice). It appears that the same set of the equations is applicable to the failure of all three types of ice. The possibility of the existence of a universal and general failure criterion for all types of ice is discussed. The validation of the present multi-surface failure criterion was discussed on the basis of comparisons between predicted failure curves and actual true triaxial test results. An overall discrepancy of predicted versus measured strength values of less than 20% was calculated. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Failure criterion; Failure envelope; Multi-surface failure; Strength; Yield; True triaxial; Sea ice; Fresh water ice; Iceberg ice
* Tel.: +1-709-772-7960; fax: +1-709-772-2462. E-mail address: [email protected] (A. Derradji-Aouat). 0165-232X/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-232X(02)00093-9
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1. Introduction and background Previously, the formulation of a unified 3-D failure criterion for isotropic fresh water ice was presented (see Derradji-Aouat, 2000). In that paper, the term unified was used to refer to both polycrystalline fresh water ice and iceberg ice (since both types of ice are isotropic, and they are made up of freshwater). The term envelope was used to indicate the ability of that failure formulation ‘‘failure criterion’’ to account for any stress path in the 3-D stress space. In the field of marine and ice engineering, the need for a 3-D failure criterion for ice stems from the fact that, during the interactions of ice with marine systems (fixed and floating offshore structures and ships), ice at the interface undergoes a complex state of stress and deformation before it fails and breaks away. This dictates that the use of the uniaxial strength of ice to compute maximum ice loads on marine systems may lead to inaccurate calculations, and subsequently, nonconclusive results. Historically, the uniaxial strength of ice has been used in empirical formulae and analytical equations to calculate ice loads on marine structures. In recent years, with the availability of high power computers (and a relatively non-expensive and easy access to HPC), numerical methods are being used more than ever before. Computer simulations based on computational techniques such as finite elements, boundary elements and discrete elements require constitutive models and failure criteria to represent the mechanical behavior and failure of ice (Derradji-Aouat, 2001). At lower strain rates ( < 10 3 s 1), the mechanical behavior of ice is nonlinear. Ice undergoes timedependent elasto-plastic deformations, micro-cracking activity, and ductile mode of failure (Derradji-Aouat, 2000). The term elasto-plastic refers to the fact that, upon loading, ice undergoes both elastic (recoverable) and plastic (permanent) deformations simultaneously. However, at high strain rates (>10 3 s 1), the mechanical behavior of ice is linear elastic with a brittle mode of failure. It seems that a range of strain rate (around c 10 3 s 1) represents a transition between the two types of ice behavior and its failure (transition from ductile to brittle mode of failure, and transition from an elasto-plastic behaviour to a linear elastic one—more details are given by Derradji-Aouat and Evgin, 2001).
The work presented in this paper deals only with the behavior and failure of ice at high strain rates (>10 3 s 1). Therefore, in this study, the stress – strain behavior of ice is considered to be linear elastic with a brittle mode of failure. Derradji-Aouat (2000) indicated that a failure envelope for fresh water isotropic ice could be formulated using the multi-surface failure theory of Mroz (1967). He used three different data sets to develop the necessary equations needed for his failure criterion. These are: two triaxial data sets on fresh water isotropic ice given by Jones (1982) and Rist and Murrell (1994) and one triaxial data set on iceberg ice given by Gagnon and Gammon (1995). Jones (1982) conducted a series of triaxial tests on cylindrical samples of isotropic fresh water ice for strain rates ranging from 1.4 10 6 to 1.4 10 2 s 1, and for confining pressures ranging from 0.1 to 85.0 MPa (0.1 MPa corresponds to the unconfined axial compressive tests). All of Jones’ tests were performed at temperature of about 12 jC. Rist and Murrell (1994) presented the results of a series of triaxial tests conducted on cylindrical isotropic fresh water ice samples. The tests were conducted at various strain rates (range from 10 5 to 10 2 s 1), for confining pressures ranging from 0.1 to 46 MPa, and for a range of temperatures from 5.2 to 44.7 jC. The combined works of Jones (1982) and Rist and Murrell (1994) provided a data set for a large range of confining pressures (0.1 to 85.0 MPa), a wide range of temperature ( 5.2 to 44.7 jC), and a large range of strain rates (1.4 10 6 to 1.4 10 2 s 1). This database enabled the development of a failure criterion that is function of the strain rate, temperature and confining pressure. Derradji-Aouat (2000) reported that the analyses of the combined data of Jones (1982) and Rist and Murrell (1994) show that the failure ‘‘strength’’ of isotropic fresh water ice can be modeled using a set of concentric elliptical curves in the triaxial plane ( q – P stress space), an example is given in Fig. 2a. The equation for the elliptical failure envelope is given as: qg 2 Pk 2 þ ¼1 ð1aÞ qmax Pc where g and k are the coordinates for the center of the ellipse, and Pc and qmax, are the major and minor axes
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of the ellipse, respectively. Table 1 gives the values g and k and Pc. The physical meaning ‘‘mechanical interpretation’’ of qmax, is that it represents the absolute maximum octahedral shear stress, it is the apex of the ellipse. Derradji-Aouat (2000) provided the following equation: qmax ¼
1=n e˙ n
ð1bÞ
where, 1 1 n ¼ 4; n ¼ 5 106 exp 10:5 103 : T 273 Eq. (1b) gives the relationship between the maximum shear stress ( qmax), strain rate (e), and temperature (T). In terms of the original multi-surface plasticity theory (Mroz, 1967), Eq. (1b) accounts for the effects of strain rate and temperature on the size of the ellipse. In terms of solid mechanics, Eq. (1b) accounts for the effects of strain rate and temperature on the maximum octahedral shear sustained by the ice ( qmax). Eq. (1b) dictates that the value of qmax (and subsequently, the size of the ellipse) increases as the strain rate increases and/or as the temperature decreases; and it decreases as the strain rate decreases and/or as the temperature increases (see an example using Jones’, 1982 data in Fig. 2a). Eqs. (1a) and (1b) were, entirely, developed from the isotropic fresh water ice test data (data of Jones, 1982; Rist and Murrell, 1994). The same equation was used to predict the results of triaxial tests on iceberg ice given by Gagnon and Gammon (1995). The main objective of these predictions was to investigate whether or not the same failure criterion ‘‘equations’’ can be used for both fresh water isotropic ice and iceberg ice. Table 1 Parameters for the elliptical failure envelope Coordinates center of the ellipse (MPa) (ellipse in the triaxial plane, q – P ) Freshwater ice and iceberg ice g
LGSIa
Major axis (MPa) Freshwater ice and iceberg ice
gs
ks
Pc
0.0
45.0
55.0
LGSI Psc
k
0.0
45.0 a
LGSI = Laboratory Grown Saline Ice.
45.0
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The triaxial data set for iceberg ice given by Gagnon and Gammon (1995) was used. Ice samples were collected from a grounded iceberg in Labrador, Canada. Cylindrical samples were tested at various strain rates (4 10 5 to 2.7 10 1 s 1) and for confining pressures ranging from 0.1 to 13.79 MPa. The tests were conducted at 1, 6, 11, and 16 jC. Derradji-Aouat (2000) reported that the analysis of Gagnon and Gammon (1995) test results showed that the failure envelope for iceberg ice could be modeled using a set of concentric elliptical curves in the triaxial plane ( q– P stress space), an example is given in Fig. 2b. Subsequently, Derradji-Aouat (2000) concluded that the same equations (and same values for their parameters) are applicable to both types of ice (fresh water isotropic ice and iceberg ice). While that study was a preliminary step towards a more comprehensive 3-D failure criterion, the overwhelming conclusion was that both isotropic fresh water ice and iceberg ice can be modeled using the same set of equations without any modifications or calibrations of their parameters. Eqs. (1a) and (1b) have been developed entirely from the results of triaxial tests (where r22 = r33, see Fig. 1). For true 3-D loading scenarios (where r11 p r22 p r33), the elliptical surface is generalized into an ellipsoid (Figs. 2 and 3); with its axis of revolution is the hydrostatic pressure (Derradji-Aouat, 2000): In 3D, the failure criterion becomes: F1 *J2D þ F2 *I2 þ F3 *ðI1 Þ2 1 ¼ 0
ð2Þ
where J2D, I1 and I2 are stresses (as defined in the nomenclature section) and F1, F2 and F3 are parameters: 8 F1 ; ; F2 ¼ 4 3ðqmax Þ2 " # 1 1 2 F3 ¼ : 3 ½Pc 2 3½qmax 2 F1 ¼
and
The main goal of the present paper is to investigate whether or not the elliptical failure formulation (Eqs. (1a) –(2)) can be extended to include saline ice (salt water ice). Mroz’s (1967) multi-surface failure theory will be used to produce the necessary equations for the criterion. If successful, the elliptical failure formulation may be viewed as a universal failure crite-
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Fig. 1. (a) A schematic for 3-D loading directions (true triaxial condition, r11 p r22 p r33). (b) A schematic for triaxial loading directions (triaxial conditions r22 = r33).
rion for all three types of ice (fresh water isotropic ice, iceberg ice, and saline ice).
2. Why multi-surface theory? Literature background In constitutive modeling of the stress-deformation characteristics of materials, the representation of the failure envelope ‘‘criterion’’ as a series of surfaces (either elliptical surface or circular surfaces in the triaxial stress plane q – P stress plot) is not new. Initially, the concept was developed by Mroz (1967)
to model yielding and calculates plastic (permanent) strains in metals. Prevost (1978) used Mroz’s (1967) concept to develop a multi-surface constitutive model for soils (sand and clay). Derradji-Aouat (1988) performed an independent validation of the applicability of Prevost’s model to granular materials (sands). Derradji-Aouat (2000) used the multi-surface concept to model the brittle failure of isotropic fresh water ice and iceberg ice. In this paper, the multisurface failure theory is extended to include the brittle failure of columnar saline ice (sea ice is mainly columnar). Columnar structure means that the material is transverse anisotropic (also known as crossisotropic material, which means isotropy in the plane perpendicular to the columns). The traditional yield and failure criteria, such as von Mises, Tresca, Drucker-Prager, and Mohr Coulomb are based on a single yield surface formulation (see schematics in Fig. 4). For clarity, it should be pointed out that, in the classical theory of plasticity (Hill, 1950), the yield surface is not a failure surface. A yield surface is the criterion for the onset of the permanent deformation (plastic flow), while a failure surface is the criterion for the maximum stresses that can be sustained by the material (strength of the material). For brittle materials with no plastic deformation (such as in the case of ice behavior at high strain rates), the yield surface becomes coincident with the failure surface. At high strain rates, in ice, once yielding is reached, ice fails (by fracture, shear, splitting, disintegration, and/or structural deterioration, etc.). It must be recognized that all of the traditional yield and failure criteria are developed for time- and temperature-independent isotropic materials (Hill, 1950). This is a significant limitation of the classical theory of plasticity when it is used to model the behavior and failure of sea ice. The latter is transverse anisotropic material,1 and its mechanical behavior depends heavily on temperature and time of loading (strain rate or stress rate dependency). Derradji-Aouat (1992) modified the classical theory of plasticity of Hill (1950) to include the effects of time (stress and strain rate) and temperature on the plastic flow, damage, and failure of columnar ice. One major advantage of the multi-surface theory is in its ability to deal with both isotropic and anisotropic 1
The vast majority of sea ice is columnar.
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Fig. 2. (a) Predicted elliptical failure envelops through Jones’ (1982) triaxial data on isotropic fresh water ice (Derradji-Aouat, 2000). (b) Predicted elliptical failure envelops through Gagnon and Gammon (1995) triaxial data on iceberg ice (Derradji-Aouat, 2000).
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Fig. 3. (a) Ellipsoidal failure envelope for fresh-water ice and iceberg ice (Derradji-Aouat, 2000). (b) A 2-D projection of the ellipsoidal failure envelope onto biaxial stress space (r1 versus r3 stress plane). (c) A section through the ellipsoidal failure envelope to show the effect of the strain rate. (d) A section through the ellipsoidal failure envelope to show the effect of temperature.
materials. This is achieved through isotropic hardening (uniform expansion/contraction of the yield surface) and/or kinematic hardening (movement of the yield surface in the stress space). An example for a multi-surface yield constitutive model and its isotropic and kinematic hardenings are given in Fig. 5a and b (for more information, see Desai and Siriwardane, 1984). Note that the expression isotropic hardening means that the ellipse expands and shrinks uniformly (the center of the ellipse is at a fixed coordinates, and the size of the axes increase or decrease as function of
the plastic deformations). The term Kinematic hardening means that the size of the ellipse is fixed, but its location changes (the axes of the ellipse are fixed, but the center moves in the stress space as a function of the plastic deformations). The same definitions ‘‘expansion and/or movement’’ are applied to circular surfaces as well as elliptical ones. Note that in the literature, combined isotropic– kinematic hardening equations (expansion and movement at the same time) have been used to model the behavior of various metals and soils.
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Fig. 4. Schematics for the traditional failure envelops.
In this paper, the idea of the multi-surface failure theory, and its fundamental principles for isotropic and kinematic hardenings, is used to model the effects of material anisotropy, temperature, and strain rate (time dependency) on the strength ‘‘failure’’ of saline ice.
3. Mechanical behavior of sea ice versus that of the fresh water ice Fundamentally, the rheology (the study of deformation and flow) of sea ice is not much different than that of the fresh water ice. The main constitutive differences between sea ice and isotropic fresh water ice are: 1. Porosity (brine volume and air volume)2 2. Material anisotropy (sea ice is generally columnar), 3. Geo-material natural variability.
total volume (or bulk volume). The brine volume is function of the ice salinity and temperature. Cox and Weeks (1983) gave all of the necessary equations needed to calculate the brine volume, the air volume, and porosity. Material Anisotropy stems from the fact that sea ice is generally columnar (some exceptions exist, such as snow sea ice and ridge sea ice). As a material, sea ice is a cross isotropic (or transverse anisotropic). When loaded across the columns, the deformation of sea ice is much different than its deformation when loaded along the columns. Geo-material Natural Variability is a generic expression that is used in this work to refer to the characteristics of sea ice in its natural environment and operating conditions. It includes the following variables:
Porosity is the relative volume of non-ice components (brine and air volume) expressed in % of the 2 In this study, the effect of solid salts is ignored (as recommended by Cox and Weeks, 1983).
The age of the sea ice (first-year sea ice, secondyear sea ice, and multi-year sea ice) The location ‘‘or source’’ (Beaufort Sea Ice, Barents Sea Ice, . . .etc.) Spatial distribution of the pores (distribution of brine and air volumes in the ice mass, whether or
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Fig. 5. (a) Multi-surface representation of the stress strain curve of a material in the 3-D stress space (Deasi and Siriardane, 1984). (b) Multisurface representation of the stress strain curve in the triaxial (q – P) plane (Deasi and Siriardane, 1984). (c) A schematic for the effect of the stress path on failure in the triaxial (q – P) plane. Starting from an initial hydrostatic pressure, P0, the elliptical failure envelope could be reached via various loading paths, such as path P 1, . . .path P n, P 5. (d) Expansion of the failure envelope as the strain rate increases and/or as the temperature decreases (the inverse is true).
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Fig. 5 (continued ).
not the pores are uniformly distributed) and the statistics for the dominating pore shapes (spheres, ellipsoids, . . .etc.) and their preferred orientation (pores main orientation is along the columns, across the columns, oblique, random, . . .etc.) The loading ‘‘stress’’ history and current physical state of the sea ice. This includes the effects of
residual loads and residual deformation, effects of open cracks, micro-cracks, effects of refrozen fractures, . . .etc.) Other. In this paper, it is recognized that the study of the effects of the ‘‘Geo-material Natural Variability’’ of
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sea ice on its strength (failure) is a major and a very complex undertaking. Therefore, at this point in time, this work will be limited only to Laboratory Grown Saline Ice (LGSI). It is hoped that after this study, the failure formulation will be extended to include real sea ice in its natural conditions.
4. Uniaxial versus multi-axial strength data of sea ice In the literature, there exists a significant amount of experimental work on the uniaxial strength of sea ice. Many researchers have conducted laboratory programs to determine the strength of sea ice and investigate its failure modes using uniaxial compression material testing machines. Some works included detailed analyses of the effects of the strain rates and temperature on the uniaxial strength of sea ice. Literature reviews were provided by Kuehn and Schulson (1994), McDonald and Jones (1998), Schulson and Gratz (1999), and several other researchers. Also, in the literature, some biaxial testing experiments were reported. These include the work of Schulson and Buck (1995), Smith and Schulson (1994), and Schulson and Nickolayev (1995). It should be noted that the works of Assur (1958) and Weeks and Assur (1967) were among the earliest laboratory investigations concerned with the basic mechanical properties of sea ice. However, in the literature, only few triaxial (and true 3-D) laboratory investigations on the strength of sea ice and saline ice are reported. It is highly possible that this situation reflects the fact that triaxial and true 3-D testing of ice requires considerable efforts and much more complex and sophisticated laboratory equipment as compared to the uniaxial testing requirements. Over the last two decades, the results of several triaxial and 3-D tests using either sea ice or saline ice were reported. These include the works of Ha¨usler (1981), Nawar et al. (1983), Timco and Frederking (1986), Sammonds et al. (1998), Sammonds and Murrell (1989), Gratz and Schulson (1994), Melton and Schulson (1995, 1998), Gratz (1996), Gratz and Schulson (1997), and Schulson and Gratz (1999). All of these investigations can be, broadly speaking, divided into two categories. The first category includes tests conducted using actual sea ice, while
the second category includes tests using Laboratory Grown Saline Ice (LGSI). Detailed information regarding the development and history of the ‘‘LGSI’’ techniques were provided by Gratz and Schulson (1994) and Gratz (1996). The subject of constitutive modeling and development of failure criteria for sea ice is a complex one. Its complexity is due, mainly, to one factor ‘‘the geomaterial natural variability’’. Consequently, in this paper, it was decided to focus the efforts on LGSI. The use of LGSI eliminates the complexity induced by the ‘‘Geo-material natural variability’’ parameters, but it allows considering the effects of salinity and porosity on the strength of ice. It is hypothesized that once an adequate validation of the present failure criterion for LGSI ice is achieved; the present equations will be extended to include the strength and failure of the actual sea ice (using the triaxial test results given by Sammonds et al., 1998 as a first step).
5. Saline ice data base The results of various true triaxial (3-D) tests given Gratz and Schulson (1994) and Gratz (1996) are used to develop the present failure criterion for saline ice. Gratz and Schulson (1994) and Gratz (1996) presented the results of a series of tests using LGSI. Pucks (diameter = 0.94 m and thickness = 0.3 m) of columnar saline ice were grown from a solution of filtered tap water mixed with aquarium salts ‘‘commercially available salt bags’’. The final ice had an average salinity of 4.5x, an average brine volume of 2.4%, and an average air volume of about 1.5%. Cubical saline ice samples (average 154 – 159 mm edge) were tested in a true triaxial machine. All tests were conducted at temperature of 10 jC and at strain rate of f 6 10 3 s 1. During each test, the applied load was controlled in one direction, and the loads in the other two directions were set as fractions of the controlled one. Fig. 1a shows a schematic for the three directions of loading (directions 1 and 2 correspond to cross columns loading, while direction 3 corresponds to along the columns loading). The test matrix included various load ratios; a load ratio is defined such that the load ratio in the controlled direction is set to 1, while the load ratios in the other two ‘‘2’’ directions are input as fractions of 1. For
A. Derradji-Aouat / Cold Regions Science and Technology 36 (2003) 47–70 Table 2 Test series #1, #2, and #3 Test Series
Series #1
Series #2
Series #3
Load ratiosa
R1; R2; R3 1; H; R 1; H; H R; R; 1 0; 0; 1 1; R; R 1; L; R R; 0; 1
R1; R2; R3 1; L; R 1; R; L 0; 0; 1 1; L; L
R1; R2; R3 1; R; L R; L; 1
a
The expressions ‘‘load ratio’’ and ‘‘load category’’ are used interchangeably. (R1; R2; R3) = stress ratios corresponding to r11, r22, r33 (see Gratz, 1996), R = load ratio (1>R>0), H = high value of R (1>H>0.9), L = low value of R (0.5>L>0), 1 = the controlled is in the corresponding direction, 0 = no load is applied in that direction.
example, if the controlled load is in direction 1, the load ratios are (1:R21:R31), where R21 and R31 are load fractions (load 2/load 1) and (load 3/load 1), respectively. Similarly, if the controlled load is in direction 2, the load ratios are (R12:1:R32), where R12 and R32 are load fractions (load 1/load 2) and (load 3/load 2), respectively. Also, if the controlled load is in direction 3, the load ratios are (R13:R23:1),
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where R13 and R23 are load fractions (load 1/load 3) and (load 2/load 3), respectively. Gratz (1996) presented the results of a large test matrix (a total of about 170 tests). It included tests for various controlled load directions (controlled load in directions 1, 2 or 3), tests for various load ratios (all three load ratios were varied between 0 and 1), and tests for various combinations of load directions and load ratios. Table 2 gives a summary of Gratz (1996) test matrix. As indicated in Table 2, depending on the load ratios, the tests were divided into three series, and each series is subdivided into several categories. There are seven categories in series #1, four categories in series #2, and two categories in series #3. In this work, it was observed that the results of tests with different controlled load directions gave valuable insights on the effect of the material anisotropy on the strength of LGSI, while tests with different load ratios gave valuable information regarding the stress path dependency. The considerations of the effect of material anisotropy on the strength of columnar ice accounts for the fact that ice samples with controlled loads along the columns have different strength ‘‘failure’’ values than those with controlled loads across the columns.
Fig. 6. Series #1 of Gratz (1996) experimental data.
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Fig. 7. (a) Series #1—experimental data versus predicted elliptical curves (load ratio: 1:1:R). (b) Series #1—Experimental data versus predicted elliptical curves (load ratio: 1:1:1). (c) Series #1—Experimental data versus predicted elliptical curves (load ratio: R:R:1). (d) Series #1— experimental data versus predicted elliptical curves (load ratio: 0:0:1). (e) Series #1—experimental data versus predicted elliptical curves (load ratio: 1:R:R). (f) Series #1—experimental data versus predicted elliptical curves (load ratio: 1:0:R). (g) Series #1—experimental data versus predicted elliptical curves (load ratio: R:0:1).
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Fig. 7 (continued ).
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Fig. 7 (continued ).
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Fig. 7 (continued ).
The stress path dependency accounts for the fact that the same strength value can be reached via various combinations of load paths (schematic for the stress path dependency is given in Fig. 5c). Although all tests were conducted at one temperature ( 10 jC) and at one strain rate value (~ 6 10 3 s 1), it will be shown (in this paper) that the previous formulation of the dependency ‘‘variation’’ of the multi-surface failure envelope on temperature and strain rate (Eq. (1b)) for fresh water isotropic ice and iceberg ice is adequate ‘‘enough’’ to include the effects of temperature and strain rates on the strength of saline ice. The isotropic hardening rule (expansion and contraction of the elliptical surface) allows extrapolating the failure envelope to any other temperature value and/or to any other strain rate values (a schematic is given in Fig. 5d).
6. Failure criterion for saline ice The analysis of the results of Gratz and Schulson (1994) and Gratz (1996) data indicated that the failure stresses of ‘‘LGSI’’ follow an elliptical curve in the
triaxial plane stress space ( q –P plane). For saline ice, the elliptical failure curve is described by an equation that is similar to Eqs. (1a) and (1b).
qs gs qsmax
2 Ps ks 2 þ ¼1 Psc
ð3Þ
where the constants gs and ks are coordinates for the center of the ellipse, qs_max is its minor axis and Psc is its major axis (numerical values given in Table 1). The stresses qs and Ps are the octahedral shear stress and the hydrostatic pressure for saline ice (they are explained below). In Eq. (3), the numerical values for the constants gs, ks, qs_max, and Psc (Table 1) are obtained using the same parametric methodology presented by DerradjiAouat (2000). Basically, elliptical curves are drawn through the strength data of a given test series and test category, and the geometric parameters of the ellipse (values for the center coordinates and both minor and major axes of the ellipse, gs, ks, qs_max, Psc) are determined by trial and error using a curve-fitting routine in the MATLAB software.
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Fig. 8. (a) Experimental data versus predicted elliptical failure curves (all tests in series #1). (b) Same as (a), but zoom on the X-axis for clear comparisons.
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The effect of the porosity is included in the term for hydrostatic pressure ( Psc, in Eq. (3)). This is due to the fact that the hydrostatic pressure has direct effect on the pores ‘‘voids’’ within the material (i.e., it has direct effect on the porosity of the material: brine volumes and air pores): Ps ¼ Pð1 þ eÞ1=n
ð4Þ
where n = 4, and the parameter e is the total porosity: Vb Va þ ð5aÞ V V Cox and Weeks (1983) gave the equations for the relative volumes of brine and air: e¼
Vb qSi ¼ V F1 ðT Þ
ð5bÞ
Va q F2 ðT Þ ¼ 1 i þ qqSi F1 ðT Þ V q
ð5cÞ
where the parameter qi is the density of ice (not the bulk density, qii0.917 mg/m3), and the F1(T) and F2(T) are
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two equations that are function of temperature and salinity (Cox and Weeks, 1983). Eq. (4) indicates that the hydrostatic pressure, P, has a direct impact on the deformation of pores and voids. In the literature, relating the hydrostatic pressure term to the porosity term is used in many constitutive models in soil mechanics (see Desai and Siriwardane, 1984; Derradji-Aouat, 1988). The effects of the material anisotropy on the strength ‘‘failure’’ of saline ice is included in the definition of the octahedral shear stress ( qs in Eq. (3)): 3 ð6aÞ qs ¼ q SR 2 where SR is the octahedral stress ratio. It is defined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 SR ¼ ðR1 R2 Þ2 þ ðR2 R3 Þ2 þ ðR3 R1 Þ2 2 ð6bÞ where the values R1, R2, and R3 are the experimental load ratios (as given in Table 2).
Fig. 9. Series #2 and #3 of Gratz (1996) experimental data.
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Fig. 10. (a) Series #2—experimental data versus predicted elliptical envelope (load ratios:1:L:R and 1:R:L). (b) Series #2—experimental data versus predicted elliptical envelope (load ratio: 0:0:1). (c) Series #2—experimental data versus predicted elliptical curves (load ratio: 1:L:L).
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Fig. 10 (continued ).
Fig. 11. Series #3—experimental data versus predicted elliptical curves (load ratios: R:L:1 and 1:R:L).
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Fig. 12. (a) Experimental data versus predicted elliptical failure curves (all tests in series #2 and #3). (b) Same as (a), but zoom on the X-axis for clear comparisons.
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In Eq. (3), the parameter qs_max is formulated as: qmax qsmax ¼ Va 100 1þ V
ð7Þ
where qmax is a parameter for the fresh water (given in Eq. (1b)) and the average relative air porosity (Va/V) is calculated using Eq. (5c) (Cox and Weeks, 1983). Therefore, for T = 10 jC (263 K) and strain rate of 6 10 3 s 1, the value of qmax is 8.48 MPa (calculated using Eq. (1b)). Since the air relative volume is 1.5%, the value qs_max = 3.4 MPa (in this work, the value of qs_max is rounded up to 3.5 MPa). Eqs. (3) –(7) give all of the necessary mathematics needed to draw elliptical failure envelopes through any triaxial data for Laboratory Grown Saline Ice (LGSI). From the development point of view, Eqs. (3) – (7) were developed in three steps. The first two steps considered the data from test series #1, while in step 3, the data from test series #2 and #3 were used. Step #1—global triaxial plot of test series #1: The test data of series #1 is divided into seven categories, each category corresponds to a given load ratio (Table 2). Fig. 6 shows the experimental data of all test categories in series #1. The data is plotted in the qs – Ps triaxial plane (maximum octahedral shear stress, qs, versus maximum hydrostatic pressure, Ps, for saline ice). At first glance, the data appears to be sketchy (high level of scatters). However, when the results for each category are plotted separately, major trends emerged (as shown in Step #2). Step #2—individual triaxial plots of tests series #1: Fig. 7a– g are individual re-plots from Fig. 6. Each figure shows the experimental data for each load category (each load ratio). Using Eq. (3) (and the parameters in Table 1 for saline ice), elliptical curves were drawn through each data category to provide comparisons between the experimental data and the calculated ‘‘predicted’’ elliptical curves. The comparison of each test category (from Fig. 7a to g) indicates that there is a ‘‘very good’’ correlation between the measured strength data of the LGSI and the calculated elliptical failure curves. Data from all test categories were plotted together with the predicted elliptical failure curves in Fig. 8a and b to provide a global comparison for all test categories in series #1. Obviously, in itself, the figure is reasonable evidence that the multi-surface elliptical
67
failure envelope formulation is capable to produce an adequate representation of the 3-D failure of saline ice (LGSI). Overall, among all comparisons, the absolute maximum difference between calculated ‘‘predicted’’ and measured strength values is less than 20%. Step #3—triaxial plots of test series #1 and #2: Fig. 9 shows the results of all tests performed in series #2 and in series #3 (for all load categories, as indicated in Table 2, the load ratios are indicated on the figure). Similar to the results of tests in series #1 (in step 1), at first glance, it appears that the data in Fig. 9 exhibit a high level of scatter. However, using Eq. (3) (and the parameters in Table 1, for the saline ice parameters), elliptical failure curves can be drawn through all strength data in series #2, and through all data in series #3. When the results of each test, for a given load ratio, are plotted individually, elliptical failure trends emerged. Fig. 10a – c shows examples for comparisons between the experimental data and their predicted elliptical curves (for test series #2), while Fig. 11 shows examples for comparisons between the experimental results and their predicted elliptical curves for series #3. The analysis of all comparisons (Figs. 10a,b,c and 11) indicates that there is a ‘‘very good’’ agreement between the measured strength data and the calculated ‘‘predicted’’ elliptical curves). Although Figs. 10c and 11 may suggest that the elliptical failure equations overestimate ‘‘very slightly’’ the measured value, the comparisons for all test categories in series #1 and all test categories in series #2 show there exists a good agreement between predicted and measured values. The absolute maximum difference between calculated ‘‘predicted’’ and measured strength values is less than 20%. Fig. 12 shows all experimental strength data versus the predicted elliptical failure envelopes for all tests in series #2 and #3 (all categories). Obviously, the figure shows that there exists a series of failure envelopes that correlates very well with the data. Equally important, the figure shows that the choice of the elliptical-shape failure curve seems to be the appropriate one.
7. Discussion: merits and limitations of the multi-surface failure criterion (Eqs. (1a), (1b) and (3) are, practically, the same, although one equation (Eqs. (1a) and (1b)) is used for
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the failure of freshwater ice and iceberg ice while the second equation (Eqs. (3)) is used for saline ice. The only difference between the two equations is that the numerical values for their parameters are different (see values given in Table 1). The difference in the parameters reflects the effects of brine and air on the strength of LGSI. The results of the experimental strength data from all three series suggest that the failure envelope for saline ice can be obtained using the same set of elliptical equations as those used for fresh water isotropic ice and iceberg ice (previously, developed by Derradji-Aouat, 2000). In this paper, the results of Gratz and Schulson (1994) and Gratz (1996) are used. Although their data set is large, more experimental data using ‘‘LGSI’’ from other laboratories and for various strain rates, temperatures, salinities, and porosities are needed to further demonstrate the applicability of the multi-surface elliptical failure envelope to saline ice. Test results from other laboratories will, surely, increase our level of confidence (or reduce our level of uncertainty) in the experimental data. Experimental Uncertainty Analysis (EUA) of the test results will help to explain the 20% discrepancy between the measured and predicted strength data. A recent Experimental Uncertainty Analysis of strength data for model ice has shown that the total uncertainty (bias uncertainty + precision uncertainty) of 15% to 30% is frequently calculated from a typical compressive strength test results (Derradji-Aouat et al., 2002). A global observation of all strength predictions (predictions presented in this paper as well as those given by Derradji-Aouat, 2000) reveals that there exists a unified multi-surface failure criterion for all three types of ice (freshwater isotropic ice, iceberg ice, and saline ice). If the existence of such a general failure envelope is validated (using additional test results from other laboratories), the present formulation should be extended to include the failure of actual sea ice (not LGSI). If successful, this will result in the development of an all inclusive failure envelope ‘‘a universal failure envelope for all types of ice’’. At this stage, it is recognized that the 3-D strength data for sea ice, in the literature, is very limited. However, Sammonds et al. (1998) triaxial data on multi-year ice can be used, as a first step, towards the development of such ‘‘universal failure envelope’’.
Over the last half a century, the universality and generality of the multi-surface failure theory has been demonstrated in metals and in soils. It is possible to demonstrate its universality and generality in ice as well. For clarity, the term ‘‘universality’’ is used to indicate that the failure criterion ‘‘actual equations’’ is applicable to various types of ice (sea ice, freshwater ice, iceberg ice, etc.), while the term ‘‘generality’’ is used to indicate that the equations are valid for all stress paths in the 3-D stress space. The work presented in this paper (using LSGI) and the previous work using fresh water isotropic ice and iceberg ice (Derradji-Aouat, 2000) points to the direction that there is a possibility for the existence of such ‘‘universal’’ and ‘‘general’’ failure envelope that is applicable to all types of ice. A major positive ramification of the development and validation of a ‘‘universal and general failure criterion’’ is in the numerical predictions of ice loads on ships and offshore structures. For instance, in the finite element software development, only one routine for the constitutive model and failure criterion of ice is needed. Naturally, the parameters of the model dependent on the ice type; their numerical values need to be specified by the users. The universality of the failure criterion reduce the complexity of the finite elements calculations of ice loads significantly (Derradji-Aouat, 2001).
8. Nomenclature and definitions The tensor notation used in this paper is the same as that given by Desai and Siriwardane (1984):
rij and eij are the stress and strain tensors. Sij is the deviatoric stress tensor. The principal stresses are: r11, r22, and r33. The expression ‘‘confining pressure’’ refers to r33. q and P are the deviatoric shear stress and hydrostatic pressure, respectively. In terms of principal stresses, q and P are:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðr11 r22 Þ2 þ ðr22 r33 Þ2 þ ðr33 r11 Þ2 q¼ 2
P¼
1 ðr11 þ r22 þ r33 Þ 3
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J2D and soct are the second invariant of the deviatoric stress tensor and the octahedral shear stress, respectively.
q¼
pffiffiffiffiffiffiffiffiffi 3 3J2D ¼ pffiffiffi soct 2
I1 and I2 = first and second invariants of the stress tensor.
I1 ¼ ðr1 þ r2 þ r3 Þ ¼ trðrÞ
r11
I2 ¼
r21
r12
r22
þ
r22 r32
r23
r11
þ
r33 r31
r13
r33
The symbols T, Si, q, e, V, Vb, and Va are used for temperature, salinity of ice, bulk density of ice, porosity (%), bulk volume, volume of brine, and volume of air, respectively. Triaxial stress plane = a 2-D plot for q versus P. 3-D stress space = a 3-D plot made up of three axes: r11, r22, and r33. Constitutive model = stress –strain (or load-deformation) mathematical relationship. Plastic deformation = permanent deformation Yield criterion = criterion for the start of the plastic deformation. Directions of loading = a schematics for a 3-D, true triaxial loading, is given in Fig. 1a (where r11 p r22 p r33) and a schematic for a triaxial loading (where r11 = r22) is given in Fig. 1b. HPC = High Power Computers (and/or High Power Computing).
References Assur, A., 1958. Composition of sea ice and its tensile strength. Conference on sea ice, Easton, Washington, National Research Council (US). ISSN 0547-8464-598. Cox, G.F.N, Weeks, W.F., 1983. Equations for determining the gas and brine volumes in sea ice samples. Journal of Glaciology 29 (102), 306 – 316. Derradji-Aouat, A., 1988. Evaluation of Prevost’s elasto-plastic models. MASc thesis, University of Ottawa, Faculty of Engineering, Ottawa, Ontario, Canada. Derradji-Aouat, A., 1992. Mathematical modelling of monotonic
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and cyclic behaviour of columnar polycrystalline freshwater ice. PhD Thesis, University of Ottawa. Faculty of Engineering, Ottawa, Ontario, Canada. Derradji-Aouat, A., 2000. A unified failure envelope for isotropic freshwater ice and iceberg ice. ASME/OMAE-2000, Int. Conference on Offshore Mechanics and Arctic Engineering, Polar and Arctic section, New Orleans, US, PDF file # OMAE-2000P/A # 1002. Derradji-Aouat, A., 2001. On the development of the Marine Dynamics Virtual Laboratory ‘‘MDVL’’. ASME-PVP-2001, Pressure Vessels and Piping Conference. Symposium on Emerging Technologies in Fluids, Structures, and Fluid – Structure Interactions, State of the Art Methods, Atlanta, Georgia, USA, pp. 349 – 357. Derradji-Aouat, A., Evgin, E., 2001. Progressive damage mechanism for ductile failure for polycrystalline ice and its complementary brittle failure criterion. ASME/OMAE-2001, Polar and Arctic section, Rio de Janeiro, Brazil, PDF file OMAE-2001 P/A # 6002. Derradji-Aouat, A., Moores, C., Stuckless, S., 2002. Terry Fox Resistance Tests-Phase 1: The ITTC Experimental Uncertainty Analysis Initiative. IMD report # TR2002-01. Desai, C.S., Siriwardane, H.J., 1984. Constitutive Laws for Engineering Materials. Prentice-Hall, Englewood Cliffs, NJ 07632, USA. Gagnon, R.E., Gammon, P.H., 1995. Triaxial experiments on iceberg and glaciers ice. Journal of Glaciology 41 (139), 528 – 540. Gratz, E.T., 1996. Brittle failure of columnar S2 ice under triaxial compression. PhD thesis. Dartmouth College, Hanover, HN, USA. Gratz, E.T., Schulson, E.M., 1994. Preliminary observations of brittle compressive failure of columnar saline ice under triaxial loading. Annals of Glaciology 19, 22 – 38. Gratz, E.T., Schulson, E.M., 1997. Brittle failure of columnar saline ice under triaxial compression. Journal of Geophysical Research 102 (B3), 5091 – 5107. Ha¨usler, F.U., 1981. Multi-axial compressive strength tests on saline ice with brush-type loading. IAHR, Symposium on Ice, Quebec, Canada, pp. 526 – 536. Hill, R., 1950. Mathematical Theory of Plasticity. Clarendon Press, Oxford. Jones, S.J., 1982. The confined compressive strength of polycrystalline ice. Journal of Glaciology 28, 171 – 177. Kuehn, G.A., Schulson, E.M., 1994. Mechanical properties of saline ice under uniaxial compression. Annals of Glaciology 19, 39 – 48. McDonald, M., Jones, S.J., 1998. Temperature and geographical dependence of the compressive and flexural strength of first year sea-ice. National Research Council of Canada, report # NRC/ IMD-LM-1998-07. Melton, J.S., Schulson, E.M., 1995. Preliminary results on the ductile deformation of columnar saline ice under triaxial compressive loading. ASME/OMAE-1995, Polar and Arctic section, vol. 4, pp. 139 – 143. Melton, J.S., Schulson, E.M., 1998. Ductile compressive failure of saline ice under triaxial conditions. Journal of Geophysical Research 103 (C10), 759 – 766.
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Mroz, Z., 1967. On the description of anisotropic work-hardening. Journal for the Mechanics and Physics of Solids 15, 163 – 175. Nawar, A.M., Nadreau, J.P., Wang, Y.S., 1983. Triaxial compressive strength of saline ice. The 7th Int. Conference on Ports and Oceans under Arctic Conditions, POAC, Helsinki, Finland, pp. 193 – 202. Prevost, J.H., 1978. Plasticity theory of soil stress – strain relations. ASCE Journal for Geotechnical Engineering 104, 347 – 361. Rist, M.A., Murrell, S.A.F., 1994. Ice triaxial deformation and fracture. Journal of Glaciology 40, 305 – 318. Sammonds, P.R., Murrell, S.A.F., 1989. A laboratory investigation of the fracture of multi-year ice under triaxial stresses at 20 jC and 40 jC. Journal of Offshore Mechanics and Arctic Engineering, JOMAE 111, 280 – 288. Sammonds, P.R., Murrell, S.A.F., Rist, M.A., 1998. Fracture of multi-year sea ice. Journal of Geophysical Research 103 (C10), 795 – 815. Schulson, E.M., Buck, S.E., 1995. The ductile to brittle transition
and ductile failure envelopes of orthotropic ice under biaxial compression. Acta Metallurgica 43 (10), 3661 – 3668. Elsevier. Schulson, E.M., Gratz, E.T., 1999. The brittle compressive failure of orthotropic ice under triaxial loading. Acta Metallurgica 47 (3), 745 – 755. Elsevier. Schulson, E.M., Nickolayev, O.Y., 1995. Failure of columnar saline ice under biaxial compression; failure envelopes and the brittle to ductile transition. Journal of Geophysical Research 100 (B11), 383 – 400. Smith, T.R., Schulson, E.M., 1994. Brittle compressive failure of salt water under biaxial conditions. Journal of Glaciology 40 (135), 265 – 276. Timco, G.W., Frederking, R.W., 1986. Confined compressive tests; outlining the failure envelope for columnar sea ice. Cold Regions Science and Technology 12, 13 – 28. Weeks, W., Assur, A., 1967. The mechanical properties of sea-ice. US Army, CRREL, NH, report # II-C3, Sept. 1967.
Multi-surface failure criterion for saline ice in the brittle regime Ahmed Derradji-Aouat * Institute for Marine Dynamics, Memorial University Campus, National Research Council Canada, P.O. Box 12093, Station A, St. John’s, Newfoundland, Canada A1B 3T5 Received 24 April 2002; accepted 7 November 2002
Abstract In this paper, the development of a 3-D failure criterion for saline ice is presented. The need for such general 3-D failure formulation stems from the fact that, during ice – ship interactions, ice undergoes a complex state of deformation and stress before it fails and breaks away, and the use of the uniaxial strength of ice to compute impact ice loads may lead to inaccurate load calculations and non-conclusive results. In recent years, with the availability of High Power Computers (HPC), numerical methods are being used more than ever before in marine and ice engineering problems. Numerical models based on computational techniques such as finite elements, boundary elements and discrete elements require 3-D constitutive models and failure criteria to represent the behavior of the materials involved (such as the behavior of the ship structure, ice and water ‘‘fluid’’). At high-speed impacts (strain rates >10 3 s 1), ice behaves as a linear elastic material with a brittle mode of failure. Previously, Derradji-Aouat [Derradji-Aouat, A., 2000. A unified failure envelope for isotropic freshwater ice and iceberg ice. ASME/OMAE-2000, Int. Conference on Offshore Mechanics and Arctic Engineering, Polar and Arctic section, New Orleans, US, PDF file # OMAE-2000-P/A # 1002] developed a unified 3-D failure envelope for both fresh water isotropic ice and iceberg ice. In this paper, that formulation is extended to include failure of saline ice (in addition to fresh water ice and iceberg ice). The results of a significant number of true triaxial tests using Laboratory Grown Ice (LGSI) were obtained from the open literature. The results of these tests formed a database that enables the existing failure model [Derradji-Aouat, A., 2000. A unified failure envelope for isotropic freshwater ice and iceberg ice. ASME/OMAE-2000, Int. Conference on Offshore Mechanics and Arctic Engineering, Polar and Arctic section, New Orleans, US, PDF file # OMAE-2000-P/A # 1002] to be extended from the isotropic fresh water ice and iceberg ice to columnar saline ice. Mroz’s [J. Mech. Phys. Solids 15 (1967) 163] concept for the multi-surface failure theory is used in both studies (the present study, for saline ice, as well as in the previous study, for the fresh water isotropic ice and iceberg ice). It appears that the same set of the equations is applicable to the failure of all three types of ice. The possibility of the existence of a universal and general failure criterion for all types of ice is discussed. The validation of the present multi-surface failure criterion was discussed on the basis of comparisons between predicted failure curves and actual true triaxial test results. An overall discrepancy of predicted versus measured strength values of less than 20% was calculated. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Failure criterion; Failure envelope; Multi-surface failure; Strength; Yield; True triaxial; Sea ice; Fresh water ice; Iceberg ice
* Tel.: +1-709-772-7960; fax: +1-709-772-2462. E-mail address: [email protected] (A. Derradji-Aouat). 0165-232X/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-232X(02)00093-9
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1. Introduction and background Previously, the formulation of a unified 3-D failure criterion for isotropic fresh water ice was presented (see Derradji-Aouat, 2000). In that paper, the term unified was used to refer to both polycrystalline fresh water ice and iceberg ice (since both types of ice are isotropic, and they are made up of freshwater). The term envelope was used to indicate the ability of that failure formulation ‘‘failure criterion’’ to account for any stress path in the 3-D stress space. In the field of marine and ice engineering, the need for a 3-D failure criterion for ice stems from the fact that, during the interactions of ice with marine systems (fixed and floating offshore structures and ships), ice at the interface undergoes a complex state of stress and deformation before it fails and breaks away. This dictates that the use of the uniaxial strength of ice to compute maximum ice loads on marine systems may lead to inaccurate calculations, and subsequently, nonconclusive results. Historically, the uniaxial strength of ice has been used in empirical formulae and analytical equations to calculate ice loads on marine structures. In recent years, with the availability of high power computers (and a relatively non-expensive and easy access to HPC), numerical methods are being used more than ever before. Computer simulations based on computational techniques such as finite elements, boundary elements and discrete elements require constitutive models and failure criteria to represent the mechanical behavior and failure of ice (Derradji-Aouat, 2001). At lower strain rates ( < 10 3 s 1), the mechanical behavior of ice is nonlinear. Ice undergoes timedependent elasto-plastic deformations, micro-cracking activity, and ductile mode of failure (Derradji-Aouat, 2000). The term elasto-plastic refers to the fact that, upon loading, ice undergoes both elastic (recoverable) and plastic (permanent) deformations simultaneously. However, at high strain rates (>10 3 s 1), the mechanical behavior of ice is linear elastic with a brittle mode of failure. It seems that a range of strain rate (around c 10 3 s 1) represents a transition between the two types of ice behavior and its failure (transition from ductile to brittle mode of failure, and transition from an elasto-plastic behaviour to a linear elastic one—more details are given by Derradji-Aouat and Evgin, 2001).
The work presented in this paper deals only with the behavior and failure of ice at high strain rates (>10 3 s 1). Therefore, in this study, the stress – strain behavior of ice is considered to be linear elastic with a brittle mode of failure. Derradji-Aouat (2000) indicated that a failure envelope for fresh water isotropic ice could be formulated using the multi-surface failure theory of Mroz (1967). He used three different data sets to develop the necessary equations needed for his failure criterion. These are: two triaxial data sets on fresh water isotropic ice given by Jones (1982) and Rist and Murrell (1994) and one triaxial data set on iceberg ice given by Gagnon and Gammon (1995). Jones (1982) conducted a series of triaxial tests on cylindrical samples of isotropic fresh water ice for strain rates ranging from 1.4 10 6 to 1.4 10 2 s 1, and for confining pressures ranging from 0.1 to 85.0 MPa (0.1 MPa corresponds to the unconfined axial compressive tests). All of Jones’ tests were performed at temperature of about 12 jC. Rist and Murrell (1994) presented the results of a series of triaxial tests conducted on cylindrical isotropic fresh water ice samples. The tests were conducted at various strain rates (range from 10 5 to 10 2 s 1), for confining pressures ranging from 0.1 to 46 MPa, and for a range of temperatures from 5.2 to 44.7 jC. The combined works of Jones (1982) and Rist and Murrell (1994) provided a data set for a large range of confining pressures (0.1 to 85.0 MPa), a wide range of temperature ( 5.2 to 44.7 jC), and a large range of strain rates (1.4 10 6 to 1.4 10 2 s 1). This database enabled the development of a failure criterion that is function of the strain rate, temperature and confining pressure. Derradji-Aouat (2000) reported that the analyses of the combined data of Jones (1982) and Rist and Murrell (1994) show that the failure ‘‘strength’’ of isotropic fresh water ice can be modeled using a set of concentric elliptical curves in the triaxial plane ( q – P stress space), an example is given in Fig. 2a. The equation for the elliptical failure envelope is given as: qg 2 Pk 2 þ ¼1 ð1aÞ qmax Pc where g and k are the coordinates for the center of the ellipse, and Pc and qmax, are the major and minor axes
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of the ellipse, respectively. Table 1 gives the values g and k and Pc. The physical meaning ‘‘mechanical interpretation’’ of qmax, is that it represents the absolute maximum octahedral shear stress, it is the apex of the ellipse. Derradji-Aouat (2000) provided the following equation: qmax ¼
1=n e˙ n
ð1bÞ
where, 1 1 n ¼ 4; n ¼ 5 106 exp 10:5 103 : T 273 Eq. (1b) gives the relationship between the maximum shear stress ( qmax), strain rate (e), and temperature (T). In terms of the original multi-surface plasticity theory (Mroz, 1967), Eq. (1b) accounts for the effects of strain rate and temperature on the size of the ellipse. In terms of solid mechanics, Eq. (1b) accounts for the effects of strain rate and temperature on the maximum octahedral shear sustained by the ice ( qmax). Eq. (1b) dictates that the value of qmax (and subsequently, the size of the ellipse) increases as the strain rate increases and/or as the temperature decreases; and it decreases as the strain rate decreases and/or as the temperature increases (see an example using Jones’, 1982 data in Fig. 2a). Eqs. (1a) and (1b) were, entirely, developed from the isotropic fresh water ice test data (data of Jones, 1982; Rist and Murrell, 1994). The same equation was used to predict the results of triaxial tests on iceberg ice given by Gagnon and Gammon (1995). The main objective of these predictions was to investigate whether or not the same failure criterion ‘‘equations’’ can be used for both fresh water isotropic ice and iceberg ice. Table 1 Parameters for the elliptical failure envelope Coordinates center of the ellipse (MPa) (ellipse in the triaxial plane, q – P ) Freshwater ice and iceberg ice g
LGSIa
Major axis (MPa) Freshwater ice and iceberg ice
gs
ks
Pc
0.0
45.0
55.0
LGSI Psc
k
0.0
45.0 a
LGSI = Laboratory Grown Saline Ice.
45.0
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The triaxial data set for iceberg ice given by Gagnon and Gammon (1995) was used. Ice samples were collected from a grounded iceberg in Labrador, Canada. Cylindrical samples were tested at various strain rates (4 10 5 to 2.7 10 1 s 1) and for confining pressures ranging from 0.1 to 13.79 MPa. The tests were conducted at 1, 6, 11, and 16 jC. Derradji-Aouat (2000) reported that the analysis of Gagnon and Gammon (1995) test results showed that the failure envelope for iceberg ice could be modeled using a set of concentric elliptical curves in the triaxial plane ( q– P stress space), an example is given in Fig. 2b. Subsequently, Derradji-Aouat (2000) concluded that the same equations (and same values for their parameters) are applicable to both types of ice (fresh water isotropic ice and iceberg ice). While that study was a preliminary step towards a more comprehensive 3-D failure criterion, the overwhelming conclusion was that both isotropic fresh water ice and iceberg ice can be modeled using the same set of equations without any modifications or calibrations of their parameters. Eqs. (1a) and (1b) have been developed entirely from the results of triaxial tests (where r22 = r33, see Fig. 1). For true 3-D loading scenarios (where r11 p r22 p r33), the elliptical surface is generalized into an ellipsoid (Figs. 2 and 3); with its axis of revolution is the hydrostatic pressure (Derradji-Aouat, 2000): In 3D, the failure criterion becomes: F1 *J2D þ F2 *I2 þ F3 *ðI1 Þ2 1 ¼ 0
ð2Þ
where J2D, I1 and I2 are stresses (as defined in the nomenclature section) and F1, F2 and F3 are parameters: 8 F1 ; ; F2 ¼ 4 3ðqmax Þ2 " # 1 1 2 F3 ¼ : 3 ½Pc 2 3½qmax 2 F1 ¼
and
The main goal of the present paper is to investigate whether or not the elliptical failure formulation (Eqs. (1a) –(2)) can be extended to include saline ice (salt water ice). Mroz’s (1967) multi-surface failure theory will be used to produce the necessary equations for the criterion. If successful, the elliptical failure formulation may be viewed as a universal failure crite-
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Fig. 1. (a) A schematic for 3-D loading directions (true triaxial condition, r11 p r22 p r33). (b) A schematic for triaxial loading directions (triaxial conditions r22 = r33).
rion for all three types of ice (fresh water isotropic ice, iceberg ice, and saline ice).
2. Why multi-surface theory? Literature background In constitutive modeling of the stress-deformation characteristics of materials, the representation of the failure envelope ‘‘criterion’’ as a series of surfaces (either elliptical surface or circular surfaces in the triaxial stress plane q – P stress plot) is not new. Initially, the concept was developed by Mroz (1967)
to model yielding and calculates plastic (permanent) strains in metals. Prevost (1978) used Mroz’s (1967) concept to develop a multi-surface constitutive model for soils (sand and clay). Derradji-Aouat (1988) performed an independent validation of the applicability of Prevost’s model to granular materials (sands). Derradji-Aouat (2000) used the multi-surface concept to model the brittle failure of isotropic fresh water ice and iceberg ice. In this paper, the multisurface failure theory is extended to include the brittle failure of columnar saline ice (sea ice is mainly columnar). Columnar structure means that the material is transverse anisotropic (also known as crossisotropic material, which means isotropy in the plane perpendicular to the columns). The traditional yield and failure criteria, such as von Mises, Tresca, Drucker-Prager, and Mohr Coulomb are based on a single yield surface formulation (see schematics in Fig. 4). For clarity, it should be pointed out that, in the classical theory of plasticity (Hill, 1950), the yield surface is not a failure surface. A yield surface is the criterion for the onset of the permanent deformation (plastic flow), while a failure surface is the criterion for the maximum stresses that can be sustained by the material (strength of the material). For brittle materials with no plastic deformation (such as in the case of ice behavior at high strain rates), the yield surface becomes coincident with the failure surface. At high strain rates, in ice, once yielding is reached, ice fails (by fracture, shear, splitting, disintegration, and/or structural deterioration, etc.). It must be recognized that all of the traditional yield and failure criteria are developed for time- and temperature-independent isotropic materials (Hill, 1950). This is a significant limitation of the classical theory of plasticity when it is used to model the behavior and failure of sea ice. The latter is transverse anisotropic material,1 and its mechanical behavior depends heavily on temperature and time of loading (strain rate or stress rate dependency). Derradji-Aouat (1992) modified the classical theory of plasticity of Hill (1950) to include the effects of time (stress and strain rate) and temperature on the plastic flow, damage, and failure of columnar ice. One major advantage of the multi-surface theory is in its ability to deal with both isotropic and anisotropic 1
The vast majority of sea ice is columnar.
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Fig. 2. (a) Predicted elliptical failure envelops through Jones’ (1982) triaxial data on isotropic fresh water ice (Derradji-Aouat, 2000). (b) Predicted elliptical failure envelops through Gagnon and Gammon (1995) triaxial data on iceberg ice (Derradji-Aouat, 2000).
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Fig. 3. (a) Ellipsoidal failure envelope for fresh-water ice and iceberg ice (Derradji-Aouat, 2000). (b) A 2-D projection of the ellipsoidal failure envelope onto biaxial stress space (r1 versus r3 stress plane). (c) A section through the ellipsoidal failure envelope to show the effect of the strain rate. (d) A section through the ellipsoidal failure envelope to show the effect of temperature.
materials. This is achieved through isotropic hardening (uniform expansion/contraction of the yield surface) and/or kinematic hardening (movement of the yield surface in the stress space). An example for a multi-surface yield constitutive model and its isotropic and kinematic hardenings are given in Fig. 5a and b (for more information, see Desai and Siriwardane, 1984). Note that the expression isotropic hardening means that the ellipse expands and shrinks uniformly (the center of the ellipse is at a fixed coordinates, and the size of the axes increase or decrease as function of
the plastic deformations). The term Kinematic hardening means that the size of the ellipse is fixed, but its location changes (the axes of the ellipse are fixed, but the center moves in the stress space as a function of the plastic deformations). The same definitions ‘‘expansion and/or movement’’ are applied to circular surfaces as well as elliptical ones. Note that in the literature, combined isotropic– kinematic hardening equations (expansion and movement at the same time) have been used to model the behavior of various metals and soils.
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Fig. 4. Schematics for the traditional failure envelops.
In this paper, the idea of the multi-surface failure theory, and its fundamental principles for isotropic and kinematic hardenings, is used to model the effects of material anisotropy, temperature, and strain rate (time dependency) on the strength ‘‘failure’’ of saline ice.
3. Mechanical behavior of sea ice versus that of the fresh water ice Fundamentally, the rheology (the study of deformation and flow) of sea ice is not much different than that of the fresh water ice. The main constitutive differences between sea ice and isotropic fresh water ice are: 1. Porosity (brine volume and air volume)2 2. Material anisotropy (sea ice is generally columnar), 3. Geo-material natural variability.
total volume (or bulk volume). The brine volume is function of the ice salinity and temperature. Cox and Weeks (1983) gave all of the necessary equations needed to calculate the brine volume, the air volume, and porosity. Material Anisotropy stems from the fact that sea ice is generally columnar (some exceptions exist, such as snow sea ice and ridge sea ice). As a material, sea ice is a cross isotropic (or transverse anisotropic). When loaded across the columns, the deformation of sea ice is much different than its deformation when loaded along the columns. Geo-material Natural Variability is a generic expression that is used in this work to refer to the characteristics of sea ice in its natural environment and operating conditions. It includes the following variables:
Porosity is the relative volume of non-ice components (brine and air volume) expressed in % of the 2 In this study, the effect of solid salts is ignored (as recommended by Cox and Weeks, 1983).
The age of the sea ice (first-year sea ice, secondyear sea ice, and multi-year sea ice) The location ‘‘or source’’ (Beaufort Sea Ice, Barents Sea Ice, . . .etc.) Spatial distribution of the pores (distribution of brine and air volumes in the ice mass, whether or
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Fig. 5. (a) Multi-surface representation of the stress strain curve of a material in the 3-D stress space (Deasi and Siriardane, 1984). (b) Multisurface representation of the stress strain curve in the triaxial (q – P) plane (Deasi and Siriardane, 1984). (c) A schematic for the effect of the stress path on failure in the triaxial (q – P) plane. Starting from an initial hydrostatic pressure, P0, the elliptical failure envelope could be reached via various loading paths, such as path P 1, . . .path P n, P 5. (d) Expansion of the failure envelope as the strain rate increases and/or as the temperature decreases (the inverse is true).
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Fig. 5 (continued ).
not the pores are uniformly distributed) and the statistics for the dominating pore shapes (spheres, ellipsoids, . . .etc.) and their preferred orientation (pores main orientation is along the columns, across the columns, oblique, random, . . .etc.) The loading ‘‘stress’’ history and current physical state of the sea ice. This includes the effects of
residual loads and residual deformation, effects of open cracks, micro-cracks, effects of refrozen fractures, . . .etc.) Other. In this paper, it is recognized that the study of the effects of the ‘‘Geo-material Natural Variability’’ of
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sea ice on its strength (failure) is a major and a very complex undertaking. Therefore, at this point in time, this work will be limited only to Laboratory Grown Saline Ice (LGSI). It is hoped that after this study, the failure formulation will be extended to include real sea ice in its natural conditions.
4. Uniaxial versus multi-axial strength data of sea ice In the literature, there exists a significant amount of experimental work on the uniaxial strength of sea ice. Many researchers have conducted laboratory programs to determine the strength of sea ice and investigate its failure modes using uniaxial compression material testing machines. Some works included detailed analyses of the effects of the strain rates and temperature on the uniaxial strength of sea ice. Literature reviews were provided by Kuehn and Schulson (1994), McDonald and Jones (1998), Schulson and Gratz (1999), and several other researchers. Also, in the literature, some biaxial testing experiments were reported. These include the work of Schulson and Buck (1995), Smith and Schulson (1994), and Schulson and Nickolayev (1995). It should be noted that the works of Assur (1958) and Weeks and Assur (1967) were among the earliest laboratory investigations concerned with the basic mechanical properties of sea ice. However, in the literature, only few triaxial (and true 3-D) laboratory investigations on the strength of sea ice and saline ice are reported. It is highly possible that this situation reflects the fact that triaxial and true 3-D testing of ice requires considerable efforts and much more complex and sophisticated laboratory equipment as compared to the uniaxial testing requirements. Over the last two decades, the results of several triaxial and 3-D tests using either sea ice or saline ice were reported. These include the works of Ha¨usler (1981), Nawar et al. (1983), Timco and Frederking (1986), Sammonds et al. (1998), Sammonds and Murrell (1989), Gratz and Schulson (1994), Melton and Schulson (1995, 1998), Gratz (1996), Gratz and Schulson (1997), and Schulson and Gratz (1999). All of these investigations can be, broadly speaking, divided into two categories. The first category includes tests conducted using actual sea ice, while
the second category includes tests using Laboratory Grown Saline Ice (LGSI). Detailed information regarding the development and history of the ‘‘LGSI’’ techniques were provided by Gratz and Schulson (1994) and Gratz (1996). The subject of constitutive modeling and development of failure criteria for sea ice is a complex one. Its complexity is due, mainly, to one factor ‘‘the geomaterial natural variability’’. Consequently, in this paper, it was decided to focus the efforts on LGSI. The use of LGSI eliminates the complexity induced by the ‘‘Geo-material natural variability’’ parameters, but it allows considering the effects of salinity and porosity on the strength of ice. It is hypothesized that once an adequate validation of the present failure criterion for LGSI ice is achieved; the present equations will be extended to include the strength and failure of the actual sea ice (using the triaxial test results given by Sammonds et al., 1998 as a first step).
5. Saline ice data base The results of various true triaxial (3-D) tests given Gratz and Schulson (1994) and Gratz (1996) are used to develop the present failure criterion for saline ice. Gratz and Schulson (1994) and Gratz (1996) presented the results of a series of tests using LGSI. Pucks (diameter = 0.94 m and thickness = 0.3 m) of columnar saline ice were grown from a solution of filtered tap water mixed with aquarium salts ‘‘commercially available salt bags’’. The final ice had an average salinity of 4.5x, an average brine volume of 2.4%, and an average air volume of about 1.5%. Cubical saline ice samples (average 154 – 159 mm edge) were tested in a true triaxial machine. All tests were conducted at temperature of 10 jC and at strain rate of f 6 10 3 s 1. During each test, the applied load was controlled in one direction, and the loads in the other two directions were set as fractions of the controlled one. Fig. 1a shows a schematic for the three directions of loading (directions 1 and 2 correspond to cross columns loading, while direction 3 corresponds to along the columns loading). The test matrix included various load ratios; a load ratio is defined such that the load ratio in the controlled direction is set to 1, while the load ratios in the other two ‘‘2’’ directions are input as fractions of 1. For
A. Derradji-Aouat / Cold Regions Science and Technology 36 (2003) 47–70 Table 2 Test series #1, #2, and #3 Test Series
Series #1
Series #2
Series #3
Load ratiosa
R1; R2; R3 1; H; R 1; H; H R; R; 1 0; 0; 1 1; R; R 1; L; R R; 0; 1
R1; R2; R3 1; L; R 1; R; L 0; 0; 1 1; L; L
R1; R2; R3 1; R; L R; L; 1
a
The expressions ‘‘load ratio’’ and ‘‘load category’’ are used interchangeably. (R1; R2; R3) = stress ratios corresponding to r11, r22, r33 (see Gratz, 1996), R = load ratio (1>R>0), H = high value of R (1>H>0.9), L = low value of R (0.5>L>0), 1 = the controlled is in the corresponding direction, 0 = no load is applied in that direction.
example, if the controlled load is in direction 1, the load ratios are (1:R21:R31), where R21 and R31 are load fractions (load 2/load 1) and (load 3/load 1), respectively. Similarly, if the controlled load is in direction 2, the load ratios are (R12:1:R32), where R12 and R32 are load fractions (load 1/load 2) and (load 3/load 2), respectively. Also, if the controlled load is in direction 3, the load ratios are (R13:R23:1),
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where R13 and R23 are load fractions (load 1/load 3) and (load 2/load 3), respectively. Gratz (1996) presented the results of a large test matrix (a total of about 170 tests). It included tests for various controlled load directions (controlled load in directions 1, 2 or 3), tests for various load ratios (all three load ratios were varied between 0 and 1), and tests for various combinations of load directions and load ratios. Table 2 gives a summary of Gratz (1996) test matrix. As indicated in Table 2, depending on the load ratios, the tests were divided into three series, and each series is subdivided into several categories. There are seven categories in series #1, four categories in series #2, and two categories in series #3. In this work, it was observed that the results of tests with different controlled load directions gave valuable insights on the effect of the material anisotropy on the strength of LGSI, while tests with different load ratios gave valuable information regarding the stress path dependency. The considerations of the effect of material anisotropy on the strength of columnar ice accounts for the fact that ice samples with controlled loads along the columns have different strength ‘‘failure’’ values than those with controlled loads across the columns.
Fig. 6. Series #1 of Gratz (1996) experimental data.
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Fig. 7. (a) Series #1—experimental data versus predicted elliptical curves (load ratio: 1:1:R). (b) Series #1—Experimental data versus predicted elliptical curves (load ratio: 1:1:1). (c) Series #1—Experimental data versus predicted elliptical curves (load ratio: R:R:1). (d) Series #1— experimental data versus predicted elliptical curves (load ratio: 0:0:1). (e) Series #1—experimental data versus predicted elliptical curves (load ratio: 1:R:R). (f) Series #1—experimental data versus predicted elliptical curves (load ratio: 1:0:R). (g) Series #1—experimental data versus predicted elliptical curves (load ratio: R:0:1).
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Fig. 7 (continued ).
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Fig. 7 (continued ).
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Fig. 7 (continued ).
The stress path dependency accounts for the fact that the same strength value can be reached via various combinations of load paths (schematic for the stress path dependency is given in Fig. 5c). Although all tests were conducted at one temperature ( 10 jC) and at one strain rate value (~ 6 10 3 s 1), it will be shown (in this paper) that the previous formulation of the dependency ‘‘variation’’ of the multi-surface failure envelope on temperature and strain rate (Eq. (1b)) for fresh water isotropic ice and iceberg ice is adequate ‘‘enough’’ to include the effects of temperature and strain rates on the strength of saline ice. The isotropic hardening rule (expansion and contraction of the elliptical surface) allows extrapolating the failure envelope to any other temperature value and/or to any other strain rate values (a schematic is given in Fig. 5d).
6. Failure criterion for saline ice The analysis of the results of Gratz and Schulson (1994) and Gratz (1996) data indicated that the failure stresses of ‘‘LGSI’’ follow an elliptical curve in the
triaxial plane stress space ( q –P plane). For saline ice, the elliptical failure curve is described by an equation that is similar to Eqs. (1a) and (1b).
qs gs qsmax
2 Ps ks 2 þ ¼1 Psc
ð3Þ
where the constants gs and ks are coordinates for the center of the ellipse, qs_max is its minor axis and Psc is its major axis (numerical values given in Table 1). The stresses qs and Ps are the octahedral shear stress and the hydrostatic pressure for saline ice (they are explained below). In Eq. (3), the numerical values for the constants gs, ks, qs_max, and Psc (Table 1) are obtained using the same parametric methodology presented by DerradjiAouat (2000). Basically, elliptical curves are drawn through the strength data of a given test series and test category, and the geometric parameters of the ellipse (values for the center coordinates and both minor and major axes of the ellipse, gs, ks, qs_max, Psc) are determined by trial and error using a curve-fitting routine in the MATLAB software.
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Fig. 8. (a) Experimental data versus predicted elliptical failure curves (all tests in series #1). (b) Same as (a), but zoom on the X-axis for clear comparisons.
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The effect of the porosity is included in the term for hydrostatic pressure ( Psc, in Eq. (3)). This is due to the fact that the hydrostatic pressure has direct effect on the pores ‘‘voids’’ within the material (i.e., it has direct effect on the porosity of the material: brine volumes and air pores): Ps ¼ Pð1 þ eÞ1=n
ð4Þ
where n = 4, and the parameter e is the total porosity: Vb Va þ ð5aÞ V V Cox and Weeks (1983) gave the equations for the relative volumes of brine and air: e¼
Vb qSi ¼ V F1 ðT Þ
ð5bÞ
Va q F2 ðT Þ ¼ 1 i þ qqSi F1 ðT Þ V q
ð5cÞ
where the parameter qi is the density of ice (not the bulk density, qii0.917 mg/m3), and the F1(T) and F2(T) are
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two equations that are function of temperature and salinity (Cox and Weeks, 1983). Eq. (4) indicates that the hydrostatic pressure, P, has a direct impact on the deformation of pores and voids. In the literature, relating the hydrostatic pressure term to the porosity term is used in many constitutive models in soil mechanics (see Desai and Siriwardane, 1984; Derradji-Aouat, 1988). The effects of the material anisotropy on the strength ‘‘failure’’ of saline ice is included in the definition of the octahedral shear stress ( qs in Eq. (3)): 3 ð6aÞ qs ¼ q SR 2 where SR is the octahedral stress ratio. It is defined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 SR ¼ ðR1 R2 Þ2 þ ðR2 R3 Þ2 þ ðR3 R1 Þ2 2 ð6bÞ where the values R1, R2, and R3 are the experimental load ratios (as given in Table 2).
Fig. 9. Series #2 and #3 of Gratz (1996) experimental data.
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Fig. 10. (a) Series #2—experimental data versus predicted elliptical envelope (load ratios:1:L:R and 1:R:L). (b) Series #2—experimental data versus predicted elliptical envelope (load ratio: 0:0:1). (c) Series #2—experimental data versus predicted elliptical curves (load ratio: 1:L:L).
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Fig. 10 (continued ).
Fig. 11. Series #3—experimental data versus predicted elliptical curves (load ratios: R:L:1 and 1:R:L).
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Fig. 12. (a) Experimental data versus predicted elliptical failure curves (all tests in series #2 and #3). (b) Same as (a), but zoom on the X-axis for clear comparisons.
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In Eq. (3), the parameter qs_max is formulated as: qmax qsmax ¼ Va 100 1þ V
ð7Þ
where qmax is a parameter for the fresh water (given in Eq. (1b)) and the average relative air porosity (Va/V) is calculated using Eq. (5c) (Cox and Weeks, 1983). Therefore, for T = 10 jC (263 K) and strain rate of 6 10 3 s 1, the value of qmax is 8.48 MPa (calculated using Eq. (1b)). Since the air relative volume is 1.5%, the value qs_max = 3.4 MPa (in this work, the value of qs_max is rounded up to 3.5 MPa). Eqs. (3) –(7) give all of the necessary mathematics needed to draw elliptical failure envelopes through any triaxial data for Laboratory Grown Saline Ice (LGSI). From the development point of view, Eqs. (3) – (7) were developed in three steps. The first two steps considered the data from test series #1, while in step 3, the data from test series #2 and #3 were used. Step #1—global triaxial plot of test series #1: The test data of series #1 is divided into seven categories, each category corresponds to a given load ratio (Table 2). Fig. 6 shows the experimental data of all test categories in series #1. The data is plotted in the qs – Ps triaxial plane (maximum octahedral shear stress, qs, versus maximum hydrostatic pressure, Ps, for saline ice). At first glance, the data appears to be sketchy (high level of scatters). However, when the results for each category are plotted separately, major trends emerged (as shown in Step #2). Step #2—individual triaxial plots of tests series #1: Fig. 7a– g are individual re-plots from Fig. 6. Each figure shows the experimental data for each load category (each load ratio). Using Eq. (3) (and the parameters in Table 1 for saline ice), elliptical curves were drawn through each data category to provide comparisons between the experimental data and the calculated ‘‘predicted’’ elliptical curves. The comparison of each test category (from Fig. 7a to g) indicates that there is a ‘‘very good’’ correlation between the measured strength data of the LGSI and the calculated elliptical failure curves. Data from all test categories were plotted together with the predicted elliptical failure curves in Fig. 8a and b to provide a global comparison for all test categories in series #1. Obviously, in itself, the figure is reasonable evidence that the multi-surface elliptical
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failure envelope formulation is capable to produce an adequate representation of the 3-D failure of saline ice (LGSI). Overall, among all comparisons, the absolute maximum difference between calculated ‘‘predicted’’ and measured strength values is less than 20%. Step #3—triaxial plots of test series #1 and #2: Fig. 9 shows the results of all tests performed in series #2 and in series #3 (for all load categories, as indicated in Table 2, the load ratios are indicated on the figure). Similar to the results of tests in series #1 (in step 1), at first glance, it appears that the data in Fig. 9 exhibit a high level of scatter. However, using Eq. (3) (and the parameters in Table 1, for the saline ice parameters), elliptical failure curves can be drawn through all strength data in series #2, and through all data in series #3. When the results of each test, for a given load ratio, are plotted individually, elliptical failure trends emerged. Fig. 10a – c shows examples for comparisons between the experimental data and their predicted elliptical curves (for test series #2), while Fig. 11 shows examples for comparisons between the experimental results and their predicted elliptical curves for series #3. The analysis of all comparisons (Figs. 10a,b,c and 11) indicates that there is a ‘‘very good’’ agreement between the measured strength data and the calculated ‘‘predicted’’ elliptical curves). Although Figs. 10c and 11 may suggest that the elliptical failure equations overestimate ‘‘very slightly’’ the measured value, the comparisons for all test categories in series #1 and all test categories in series #2 show there exists a good agreement between predicted and measured values. The absolute maximum difference between calculated ‘‘predicted’’ and measured strength values is less than 20%. Fig. 12 shows all experimental strength data versus the predicted elliptical failure envelopes for all tests in series #2 and #3 (all categories). Obviously, the figure shows that there exists a series of failure envelopes that correlates very well with the data. Equally important, the figure shows that the choice of the elliptical-shape failure curve seems to be the appropriate one.
7. Discussion: merits and limitations of the multi-surface failure criterion (Eqs. (1a), (1b) and (3) are, practically, the same, although one equation (Eqs. (1a) and (1b)) is used for
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the failure of freshwater ice and iceberg ice while the second equation (Eqs. (3)) is used for saline ice. The only difference between the two equations is that the numerical values for their parameters are different (see values given in Table 1). The difference in the parameters reflects the effects of brine and air on the strength of LGSI. The results of the experimental strength data from all three series suggest that the failure envelope for saline ice can be obtained using the same set of elliptical equations as those used for fresh water isotropic ice and iceberg ice (previously, developed by Derradji-Aouat, 2000). In this paper, the results of Gratz and Schulson (1994) and Gratz (1996) are used. Although their data set is large, more experimental data using ‘‘LGSI’’ from other laboratories and for various strain rates, temperatures, salinities, and porosities are needed to further demonstrate the applicability of the multi-surface elliptical failure envelope to saline ice. Test results from other laboratories will, surely, increase our level of confidence (or reduce our level of uncertainty) in the experimental data. Experimental Uncertainty Analysis (EUA) of the test results will help to explain the 20% discrepancy between the measured and predicted strength data. A recent Experimental Uncertainty Analysis of strength data for model ice has shown that the total uncertainty (bias uncertainty + precision uncertainty) of 15% to 30% is frequently calculated from a typical compressive strength test results (Derradji-Aouat et al., 2002). A global observation of all strength predictions (predictions presented in this paper as well as those given by Derradji-Aouat, 2000) reveals that there exists a unified multi-surface failure criterion for all three types of ice (freshwater isotropic ice, iceberg ice, and saline ice). If the existence of such a general failure envelope is validated (using additional test results from other laboratories), the present formulation should be extended to include the failure of actual sea ice (not LGSI). If successful, this will result in the development of an all inclusive failure envelope ‘‘a universal failure envelope for all types of ice’’. At this stage, it is recognized that the 3-D strength data for sea ice, in the literature, is very limited. However, Sammonds et al. (1998) triaxial data on multi-year ice can be used, as a first step, towards the development of such ‘‘universal failure envelope’’.
Over the last half a century, the universality and generality of the multi-surface failure theory has been demonstrated in metals and in soils. It is possible to demonstrate its universality and generality in ice as well. For clarity, the term ‘‘universality’’ is used to indicate that the failure criterion ‘‘actual equations’’ is applicable to various types of ice (sea ice, freshwater ice, iceberg ice, etc.), while the term ‘‘generality’’ is used to indicate that the equations are valid for all stress paths in the 3-D stress space. The work presented in this paper (using LSGI) and the previous work using fresh water isotropic ice and iceberg ice (Derradji-Aouat, 2000) points to the direction that there is a possibility for the existence of such ‘‘universal’’ and ‘‘general’’ failure envelope that is applicable to all types of ice. A major positive ramification of the development and validation of a ‘‘universal and general failure criterion’’ is in the numerical predictions of ice loads on ships and offshore structures. For instance, in the finite element software development, only one routine for the constitutive model and failure criterion of ice is needed. Naturally, the parameters of the model dependent on the ice type; their numerical values need to be specified by the users. The universality of the failure criterion reduce the complexity of the finite elements calculations of ice loads significantly (Derradji-Aouat, 2001).
8. Nomenclature and definitions The tensor notation used in this paper is the same as that given by Desai and Siriwardane (1984):
rij and eij are the stress and strain tensors. Sij is the deviatoric stress tensor. The principal stresses are: r11, r22, and r33. The expression ‘‘confining pressure’’ refers to r33. q and P are the deviatoric shear stress and hydrostatic pressure, respectively. In terms of principal stresses, q and P are:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðr11 r22 Þ2 þ ðr22 r33 Þ2 þ ðr33 r11 Þ2 q¼ 2
P¼
1 ðr11 þ r22 þ r33 Þ 3
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J2D and soct are the second invariant of the deviatoric stress tensor and the octahedral shear stress, respectively.
q¼
pffiffiffiffiffiffiffiffiffi 3 3J2D ¼ pffiffiffi soct 2
I1 and I2 = first and second invariants of the stress tensor.
I1 ¼ ðr1 þ r2 þ r3 Þ ¼ trðrÞ
r11
I2 ¼
r21
r12
r22
þ
r22 r32
r23
r11
þ
r33 r31
r13
r33
The symbols T, Si, q, e, V, Vb, and Va are used for temperature, salinity of ice, bulk density of ice, porosity (%), bulk volume, volume of brine, and volume of air, respectively. Triaxial stress plane = a 2-D plot for q versus P. 3-D stress space = a 3-D plot made up of three axes: r11, r22, and r33. Constitutive model = stress –strain (or load-deformation) mathematical relationship. Plastic deformation = permanent deformation Yield criterion = criterion for the start of the plastic deformation. Directions of loading = a schematics for a 3-D, true triaxial loading, is given in Fig. 1a (where r11 p r22 p r33) and a schematic for a triaxial loading (where r11 = r22) is given in Fig. 1b. HPC = High Power Computers (and/or High Power Computing).
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