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Yield and failure envelope for ice under multiaxial compressive stresses
Cold Regions Science and Technology, 13 (1986) 75 -82
75
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
Y I E L D A N D F A I L U R E E N V E L O P E FOR ICE U N D E R M U L T I A X I A L STRESSES
COMPRESSIVE
Jean-Paul Nadreau C-CORE, Memorial University of Newfoundland, St. John's, Newfoundland A IB 3)(5 (Canada)
and Bernard Michel Universit~ Laval, Quebec, Qu6bec GI K 7P4 (Canada)
(ReceivedNovember7, 1985; acceptedin revisedform April 17, 1986)
ABSTRACT
An analysis has been made on Jones' (1982) experimental results on the yieM strength of ice. The available data were obtained from triaxial tests performed on polycrystalline isotropic ice. Maximum confining pressure reached 85 MPa. These data have been analysed in order to formulate a proper yield criterion for ice (Nadreau, 1985). The model was influenced by previous studies on ice yieM criteria which were derived from equations used in the field of rock and soil mechanics. One of the main advantages of this new formulation is its ability to consider the hydrostatic pressure at which ice changes to water (Mellor, 1980). The yieM surface is a cubic function of the invariants of the stress tensors. The parameters involved in the definition of the yield envelope are functions of the hydrostatic pressure at phase change and of the strain rate applied to the sample. The three-dimensional surface defined by this criterion is teardrop-shaped and symmetrical around the hydrostatic line. The volume of this teardrop envelope is found to increase as a function of increasing strain-rate.
INTRODUCTION Probably because uniaxial compression tests are the easiest ones to do, the amount of data available on uniaxial compression is several times larger than 0165-232X/86/$03.50
that available for uniaxial tension or, even worse, for confined compression. Multiaxial data and theories have been available in rocks and softs mechanics for a long time. The need for triaxial tests, however, was not so apparent to glaciologists who were concerned with creep at low strain rates at which hydrostatic pressure has little influence on ice behaviour. It seems appropriate, when dealing with very low strain rates, to consider that deviatoric stresses only affect the behaviour of ice; the yield criterion can be identified at first approximation as Tresca's (Michel, 1978, 1981; Nadreau and Michel, 1984; Coon et al., 1984). On the other hand, a few analyses have been carried out on the available data for multiaxially stressed ice (Jones, 1978, 1982; H/iusler, 1981). These studies applied behaviour models used in the field of rock mechanics to ice. First, Reinicke and Ralston (1977) used the quadratic form of the n-type yield equation proposed by Pariseau 0968). Later, Reinicke and Reimer 0 9 7 8 ) used the formulation developed by Smith (1974) for anisotropic rocks and softs and applied it to triaxial data obtained by Jones (1978). Having been developed for rocks and softs, both Pariseau's and Smith's models fail to take into account the vanishing resistance of ice due to its phase change to water when a certain level of hydrostatic pressure is reached. Mellor (1980) was the first to suggest that in attempting to formulate a general failure criterion,
© 1986 ElsevierScience PublishersB.V.
76 the phase change due to hydrostatic pressure in conjunction with temperature should be taken into account. The model presented here gives priority to this characteristic of ice.
assumed to fail when the maximum shear stress reaches a certain value. This can be written Jtc(a) ----
(0"1-- 0'3)
-- 0"r
YIELD FUNCTIONS Yield functions are used to express the behaviour of a material mathematically. They are usually presented as an algebraic combination of the stress components 0"i/. Physically when f(0"i/) < 0, the yield functions identify a region of the three-dimensional stress space in which the material behaves as an elastic body. The surface where f(0"ii) = 0, defines the locus where the material is at yield and where it starts behaving as a plastic body. The yield functions of Tresca, yon Mises and Mohr-Coulomb are the most common criteria used to describe yield behaviour of soils and rocks. Tresca's criterion states that the material is
Fig. 1. Yield surface as defined by von Mises' criterion.
I
15 o n
~
5
U o~ In
tu
,'~
0
sssi/iI o
Experimental results (Jones,19?8) I
•-
Von Mises
Mohr-Coulomb Pariseau Smith
-I0
0
I0
i
i
I
I
I
20 30 40 Confining pressure (T3, MPa
50
Fig. 2. Comparison of yield criteria (Reinicke and Reimer, 19"/8).
I 60
70
77 where ol and 03 are the maximum and minimum principal stresses and where Or is the unconf'med compressive or tensile strength of the material. If the material is assumed to fail when the octahedral shear stress reaches a value characteristic of the material or when distortion energy is considered as the failing criterion one can write yon Mises' yield function: f(o)
& = ~ Kay - o z ) : + (Oz - Ox): + (ox - Oy)q 7"yz2 + 7"zx2 + ,rxy 2 .
A representation of this yield surface is presented in Fig. 1. Consider a point M on this surface; the stress vector ~ (o) can be separated into its two components: stress due to hydrostatic pressure (dilat tational stress) O---1 $ (oo); and deviatoric stress PM (o). The yield ellipse which lies in the o x O o y plane represents the plane stress failure envelope associated with yon Mises' criterion. Tresca's yield hexagon has been drawn for comparison. One of the most commonly used criteria is the M o h r - C o u l o m b yield function which states that confining pressure increases material strength due to the internal friction. The yield function takes the following form:
1
= ~ [(Ox - Oy) 2 + (Oy - Oz) 2 + (Oz -- Ox) 2 ] + 3 [ryz 2 + rxz 2 + rxy 2] -Or 2 .
This expression can be simplified when expressed in function of stress tensor invariants. It becomes: f(o)
= a J2
where a becomes 1~or ~ and where J2is the second invariant of the deviatoric stress tensor, given by the expression:
o
= o l (1 - sin ~) - 03 (1 + sin ¢) - 2 c(cos ~)
f(o)
25
O.
F-
b
20
15
IO
/,/;' / , / 9 / , / 9 "~--- °'z: $ ~
MEASURED STRENGTHS ( H~uster, 1981)
°!
!
l
-5
..
0
5
I
I
r
I0
I
[
I
,
I
I
15
1
I
1
20
I
I
1
25
a'x, MPa
Fig. 3. Projections of the Smith and Pariseau yield surface of saline ice (H~iusler, 1983).
78 where c and ¢ are the cohesion and the internal friction angle, respectively. Finally, two other models, designed to describe anisotropic rocks and soils, have been applied to ice. Pariseau (1968) proposed a formulation of the yield criterion which would take into account the influence of the intermediate principal stress (02). When simplified to represent isotropic material behaviour, it becomes: f(o)
= a J2 n n + bI1 - 1
where a and b are material constants, 11 the first invariant of stress, given by 11 = Ox + Oy + Oz, and J2 the second invariant of deviatoric stress. This criterion has been used by Reinicke and Ralston (1977) and Ralston (1977, 1978). They assumed n=2 and applied this function to ice indentation and plane strain analysis. A generalization of this equation was developed by Smith (t974) and applied to rock deformation. This formulation can be written for an isotropic material: f(o)
= a J2 + bI1 + c I 1 = -
ol - 03 = tan0 (C - 03) (l + (03 - C ) / ( C - 73) 1/2 where al is the maximum axial stress, 03 the confining pressure, 0 the half angle of the tail, C the phase change pressure (function of temperature), and T a biaxial tension reference stress. The parameter T, which should represent the biaxial yielding tension (al = 0 and 02 = a3 < 0), is the weakest point of the model. This value is to be considered only as a fitting parameter; it was the best available value for all the tests curves involved but does not correspond to the real tensile strength of ice. One has to keep in mind that the model presented 20
1 .
Reinicke and Reimer (1978) used this equation to analyse triaxial test data obtained by Jones (1978) and compared it to the other yield criteria mentioned above. Figure 2 reproduces the comparison they made. Since then, H~iusler (1983) has used the data he obtained from true (O 1 ::~ O"2 ¢ O'3)triaxial tests (H/iusler, 1981) to compare both Smith's and Pariseau's models. Figure 3 shows the result. YIELD CRITERION FORMULATION
account (MeUor, 1980). This function reproduces quite nicely the evolution of the deviatoric strength of ice as confining pressure increases. The ice exhibits some increase in strength up to 2 0 - 3 0 MPa and slowly decreases down to the phase change point. Using stress parameters defined in Fig. 5, the general expression of this function is written:
g
~ -20 E
FOR ICE, A NEW
As mentioned by Mellor (1980), these models were designed for rocks and soils and do not describe the feature which characterizes ice when subjected to hydrostatic pressure. Ice is known to change phases according to the phase diagram shown in Fig. 4. When dealing with pressures up to 200 MPa the liquidus decreases from O°C down to approximately -22°C. This particular feature of the ice and the data obtained by Jones (1982) from polycrystalline ice led the authors to propose a teardrop function as the best fit for these results (Nadreau, 1985). This teardrop shape satisfies the characteristics that a general failure criterion for ice should take into
"40
k•ll
\ \
I
0.2
\
V \
/I
\ \ \
I 0.4
0.6
0.8
Pressure , GPO
Fig. 4. Phase diagram for ice.
T
Fig. 5. "Teardrop" configuration.
C
79
rate is shown on Fig. 8. The evolution of the tail angle can be described by the following empirical relation:
here has been designed for multiaxial compression stress state only; since tension and compression fracture behaviours are completely different, one cannot expect a simple model to fit both fields. This model has been applied to the data presented by Jones (1982). The tests covered a wide range of strain rates from 1.4 X 10 -6 s-1 to 1.4 X 10 -2 s-a and were performed at different temperatures. The values considered as yield stresses are the maximum values encountered during the tests, which correspond, for low strain rates, to the initiation of plastic yielding and flow and, for higher strain rates, to brittle failure (Jones, 1982). The best fit curves for each strain rate series of Jones' results are presented in Fig. 6. It shows how the function covers the data perfectly. The tests were performed at -12°C. The hydrostatic pressure which makes the ice turn back to water at this temperature is approximately 130 MPa. The tension reference strength has been taken as 10 MPa for all the curves. Even if the differences between the theoretical curve and the experimental data points are magnified because of the expansion of the ordinate axis, the fitting appears quite good. When plotted in an isometric reference system the model looks quite appropriate for the multiaxial stress state (Fig. 7). It is worthwhile to note the evolution of the teardrop tail half angle with respect to strain rate. The plotting of the best fit values of 0 for each strain
0
=
K~ q
where q = 0.23 and K = 70.7 (0 being in degrees and 1 X 10 -s s -1 < ~ < 1 X 10 -2 s-X). The adequacy of this model to describe the confined compressive data leads the authors to use the teardrop shape to generate a three-dimensional yield surface for ice. In order to achieve the generalization, the isotropic behaviour of the granular ice is considered. This characteristic of the ice implies that the yield surface is symmetrical around the hydrostatic axis. The steps to generate the three dimensional surface are: first, generate the revolution surface around an axis (say, y) and then transform the equation in order to move the revolution axis from y to the hydrostatic axis. The starting equation being in the y, z plane (Fig. 9), Z
=
A t ( C - y ) O;
- -
T ) 1/2 ,
with A' = a/(C - 7') v2 and a = tan 0, the revolution surface equation can be written: x2+z 2 = A(C-y)2(y-T) withA =A '2. When using the transformation direction cosines
30
EL 0
G .
.
.
.
20
03
I0
--//A2,
°
,-.:l\\----
L.
.9 ° >
£3 0 -20
0
20
40
60
80
Confining pressure, 0-3,MPo Fig. 6. ]ones' triaxial tests.
I00
120
140
80
a 2 (x _y2) + (7(x + y ) + Gz) 2 - A (C-[3(x +y +z)) 2 X
shown in Table 1 and Fig. 10, (new coordinates being kept x, y , z in order to avoid prime superscripts), the equation becomes:
(~(x+y+z)-r)
= 0
140
120
I00
80
u Q,.
60
b--
4o
2o X
-20
- 20
0
2o
40
Confining
60
80
120
iO0
140
pressure, O'3,MPo
Fig. 7. Jones' triaxial tests. I00
7
I0
7
J i
=o o I
.
I 0 -e
.
.
.
.
.
.
i 0 -~
.
.
.
.
.
.
.
.
.
.
.
.
10.4 Strain
10.3 r a l e , s°I
Fig. 8. Evolution of the tail half-angle with respect to strain rate.
,
i
I
I I
i
10.2
I
I
i
I
I
I
il
i 0 -t
81
z
OZ
/
/
/
L._f j /
/
X
Fig. 9. Generating surface.
t
j ""
~
(7x Fig. 11. Yield surface for isotropic ice, new formulation. Fig. 10. Transformation of axes.
can be written as a function o f stress invariants; finally the model can be formulated as:
TABLE 1
f(oii)
Values of the angles and the direction cosines relating the new axes to the old set Angles x
y
z
54.7 54.7 54.7
114.1 114.1 35.3
Direction cosines x y z
= aI13 + bI12 + c i 1 + d J2 - 1 = 0
with a = 3a/TC 2 ,
b = -32(2 C + T)/TC ~ , x' y' z'
45 135 0
a --a 0
3 3 #
"r 6
If the development is carried out and the x , y , z are replaced b y a l , o~, o3 the expression o f the surface is:
A •3 (01 -k o"2 -[-0"3)3
A ~2 (2 C + T) (0"1 "t"0.2 "b0.3) 2 2 + A 3 C(2 T + C) (01 + 0.2 + 0"a) - § (02 - 0.3)2 -
-
+ (0.3-0.1) 2 + (0"1-0"2) 2 - A T C 2 = 0 This expression, being the yield surface o f a material,
c = 3(2T+C)/TC, d = -2/3A
TC 2 ,
I1 being.the first invariant o f the stress tensor and J2 the second invariant o f the deviatoric stress tensor. A representation o f this modelis presented in athree dimensional reference system in Fig. 11. It should be noted that the volume o f the teardrop defined in Fig. 11 increases as strain rate increases. Moreover, the temperature decrease makes the teardrop longer on the hydrostatic axis. These trends are valid when the temperature remains over -22°C and when strain rates range from 1 × 10 -6 s -1 to 1 × 10 -2 s -1. This model, developed for isotropic ice, could also
82 be extrapolated to describe the anisotropic behaviour of other types of ice. It would take the following form: f ( o i ] ) = al ( a y - Oz) 2 + a2 (tXz - ax) 2 + a3 (Ox - ay) 2 +
2 + a6 2 a4 J, y z + as Tzx Txy
+ a7 Ox + a8 o3, + a 9 0 z + (alo (~x + a l l ay + a12 a z ) 2
+ (a13 ax + a14 Oy + als Oz) 3 - 1 = 0 . For the complete anisotropic case fifteen coefficients would have to be determined from experimental tests. The number of coefficients would be reduced to ten if horizontal anisotropy is involved (as in $2 ice for instance).
CONCLUSION Using experimental data obtained from triaxial tests on polycrystalline ice, a model was developed which represents the behaviour of the ice when subjected to confining pressure. This model is an improvement over the existing yield criteria which failed in describing the effects of phase change due to hydrostatic pressure. The model presented here remains rather simple, and describes the ice behaviour more accurately at high confining pressures. It has been developed for isotropic ice but can be extrapolated to represent anisotropic types as well. Being based exclusively on ice data, it seems more appropriate than the other models derived from soils and rocks mechanics. It should contribute to the use of a proper yield criterion for ice in compression and should be useful to engineers.
ACKNOWLEDGEMENTS The information presented in this paper is part of a Ph.D. thesis undertaken under the supervision of Dr B. Michel at Laval University in Quebec City. This study has been possible thanks to a grant
from the Natural Sciences and Engineering Research Council of Canada.
REFERENCES Coon, M.D., Evans, R.J. and Gibson, B.H. (1984). Failure criteria for sea ice and loads resulting from crushing. Proc. IAHR 1984, Hamburg, Vol. 3, pp. 1-16. H/iusler, F.U. (1981). Multiaxial compressive strength tests on saline ice with brush-type loading platens. Proc. IAHR 1981, Quebec City, Vol. 2, pp. 526-539. H/iusler, F.U. (1983). Comparison between different yield functions for saline ice. Ann. Glaeiol., 4: 105-109. Jones, S.J. (1978). Triaxial testing of polycrystalline ice. Proc. Third Int. Conf. on Permafrost, Edmonton, Vol. 1, pp. 670-674. Jones, S.J. (1982). The confined compressive strength of polyerystalline ice. J. Glaciol., 20(98): 171-177. Mellor, M. (1980). Mechanical properties of polycrystalline ice. In Physics and Mechanics of Ice, IUTAM Symp., Copenhagen 1979, pp. 217-245. Michel, B. (1978). Ice Mechanics. Les presses de L'Universit6 Laval, Quebec City, 499 pp. Michel, B. (1981). Advances in ice mechanics. Proc. POAC 81, Quebec City, Quebec, pp. 189-204. Nadreau, J.P. (1985). Lois de comportement et de fluage de la glace granulaire simul~e de crates de pression. Ph.D. Thesis, Laval University, Quebec City, Quebec. Nadreau, J.P. and Michel, B. (1984). lee properties in relation to ice forces. Second state-of-the-art report, IAHR workhag group on ice forces. Proe. IAHR 1984, Hamburg, Vol. 4, Chap. 1. Pariseau, W.G. (1968). Plasticity theory for anisotropie rocks and soils. In K.E. Gray (ed.), Proc. Tenth Symp. on Rocks Mechanics, pp. 267-295. Ralston, T.D. (1977). Yield and plastic deformation in ice crushing failure. Proc. of the Arctic Ice dynamics joint experiment, Int. Commission on Snow and Ice Syrup., Univ. Washington Press, pp. 234-245. Ralston, T.D. (1978). An analysis of ice sheet indentation. Proc. IAHR 1978, Lulea, Sweden, Vol. 1, pp. 13-31. Reinicke, K.M. and Ralston, T.D. (1977). Plastic limit analysis with an anisotropic, parabolic yield function. Int. J. Rock Mech., Min. Sci. Geomech. Abstr., 14: 147-154. Reinicke, K.M. and Reimer, R. (1978). A procedure for the determination of ice forces - illustrated for polycrystalline ice. Proc. IAHR 1978, Lulea, Sweden, Vol. 1, pp. 217-238. Smith, M.B. (1974). A parabolic yield condition for anisotropic rocks and soils. Ph.D. thesis, Rice University, Houston.
75
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
Y I E L D A N D F A I L U R E E N V E L O P E FOR ICE U N D E R M U L T I A X I A L STRESSES
COMPRESSIVE
Jean-Paul Nadreau C-CORE, Memorial University of Newfoundland, St. John's, Newfoundland A IB 3)(5 (Canada)
and Bernard Michel Universit~ Laval, Quebec, Qu6bec GI K 7P4 (Canada)
(ReceivedNovember7, 1985; acceptedin revisedform April 17, 1986)
ABSTRACT
An analysis has been made on Jones' (1982) experimental results on the yieM strength of ice. The available data were obtained from triaxial tests performed on polycrystalline isotropic ice. Maximum confining pressure reached 85 MPa. These data have been analysed in order to formulate a proper yield criterion for ice (Nadreau, 1985). The model was influenced by previous studies on ice yieM criteria which were derived from equations used in the field of rock and soil mechanics. One of the main advantages of this new formulation is its ability to consider the hydrostatic pressure at which ice changes to water (Mellor, 1980). The yieM surface is a cubic function of the invariants of the stress tensors. The parameters involved in the definition of the yield envelope are functions of the hydrostatic pressure at phase change and of the strain rate applied to the sample. The three-dimensional surface defined by this criterion is teardrop-shaped and symmetrical around the hydrostatic line. The volume of this teardrop envelope is found to increase as a function of increasing strain-rate.
INTRODUCTION Probably because uniaxial compression tests are the easiest ones to do, the amount of data available on uniaxial compression is several times larger than 0165-232X/86/$03.50
that available for uniaxial tension or, even worse, for confined compression. Multiaxial data and theories have been available in rocks and softs mechanics for a long time. The need for triaxial tests, however, was not so apparent to glaciologists who were concerned with creep at low strain rates at which hydrostatic pressure has little influence on ice behaviour. It seems appropriate, when dealing with very low strain rates, to consider that deviatoric stresses only affect the behaviour of ice; the yield criterion can be identified at first approximation as Tresca's (Michel, 1978, 1981; Nadreau and Michel, 1984; Coon et al., 1984). On the other hand, a few analyses have been carried out on the available data for multiaxially stressed ice (Jones, 1978, 1982; H/iusler, 1981). These studies applied behaviour models used in the field of rock mechanics to ice. First, Reinicke and Ralston (1977) used the quadratic form of the n-type yield equation proposed by Pariseau 0968). Later, Reinicke and Reimer 0 9 7 8 ) used the formulation developed by Smith (1974) for anisotropic rocks and softs and applied it to triaxial data obtained by Jones (1978). Having been developed for rocks and softs, both Pariseau's and Smith's models fail to take into account the vanishing resistance of ice due to its phase change to water when a certain level of hydrostatic pressure is reached. Mellor (1980) was the first to suggest that in attempting to formulate a general failure criterion,
© 1986 ElsevierScience PublishersB.V.
76 the phase change due to hydrostatic pressure in conjunction with temperature should be taken into account. The model presented here gives priority to this characteristic of ice.
assumed to fail when the maximum shear stress reaches a certain value. This can be written Jtc(a) ----
(0"1-- 0'3)
-- 0"r
YIELD FUNCTIONS Yield functions are used to express the behaviour of a material mathematically. They are usually presented as an algebraic combination of the stress components 0"i/. Physically when f(0"i/) < 0, the yield functions identify a region of the three-dimensional stress space in which the material behaves as an elastic body. The surface where f(0"ii) = 0, defines the locus where the material is at yield and where it starts behaving as a plastic body. The yield functions of Tresca, yon Mises and Mohr-Coulomb are the most common criteria used to describe yield behaviour of soils and rocks. Tresca's criterion states that the material is
Fig. 1. Yield surface as defined by von Mises' criterion.
I
15 o n
~
5
U o~ In
tu
,'~
0
sssi/iI o
Experimental results (Jones,19?8) I
•-
Von Mises
Mohr-Coulomb Pariseau Smith
-I0
0
I0
i
i
I
I
I
20 30 40 Confining pressure (T3, MPa
50
Fig. 2. Comparison of yield criteria (Reinicke and Reimer, 19"/8).
I 60
70
77 where ol and 03 are the maximum and minimum principal stresses and where Or is the unconf'med compressive or tensile strength of the material. If the material is assumed to fail when the octahedral shear stress reaches a value characteristic of the material or when distortion energy is considered as the failing criterion one can write yon Mises' yield function: f(o)
& = ~ Kay - o z ) : + (Oz - Ox): + (ox - Oy)q 7"yz2 + 7"zx2 + ,rxy 2 .
A representation of this yield surface is presented in Fig. 1. Consider a point M on this surface; the stress vector ~ (o) can be separated into its two components: stress due to hydrostatic pressure (dilat tational stress) O---1 $ (oo); and deviatoric stress PM (o). The yield ellipse which lies in the o x O o y plane represents the plane stress failure envelope associated with yon Mises' criterion. Tresca's yield hexagon has been drawn for comparison. One of the most commonly used criteria is the M o h r - C o u l o m b yield function which states that confining pressure increases material strength due to the internal friction. The yield function takes the following form:
1
= ~ [(Ox - Oy) 2 + (Oy - Oz) 2 + (Oz -- Ox) 2 ] + 3 [ryz 2 + rxz 2 + rxy 2] -Or 2 .
This expression can be simplified when expressed in function of stress tensor invariants. It becomes: f(o)
= a J2
where a becomes 1~or ~ and where J2is the second invariant of the deviatoric stress tensor, given by the expression:
o
= o l (1 - sin ~) - 03 (1 + sin ¢) - 2 c(cos ~)
f(o)
25
O.
F-
b
20
15
IO
/,/;' / , / 9 / , / 9 "~--- °'z: $ ~
MEASURED STRENGTHS ( H~uster, 1981)
°!
!
l
-5
..
0
5
I
I
r
I0
I
[
I
,
I
I
15
1
I
1
20
I
I
1
25
a'x, MPa
Fig. 3. Projections of the Smith and Pariseau yield surface of saline ice (H~iusler, 1983).
78 where c and ¢ are the cohesion and the internal friction angle, respectively. Finally, two other models, designed to describe anisotropic rocks and soils, have been applied to ice. Pariseau (1968) proposed a formulation of the yield criterion which would take into account the influence of the intermediate principal stress (02). When simplified to represent isotropic material behaviour, it becomes: f(o)
= a J2 n n + bI1 - 1
where a and b are material constants, 11 the first invariant of stress, given by 11 = Ox + Oy + Oz, and J2 the second invariant of deviatoric stress. This criterion has been used by Reinicke and Ralston (1977) and Ralston (1977, 1978). They assumed n=2 and applied this function to ice indentation and plane strain analysis. A generalization of this equation was developed by Smith (t974) and applied to rock deformation. This formulation can be written for an isotropic material: f(o)
= a J2 + bI1 + c I 1 = -
ol - 03 = tan0 (C - 03) (l + (03 - C ) / ( C - 73) 1/2 where al is the maximum axial stress, 03 the confining pressure, 0 the half angle of the tail, C the phase change pressure (function of temperature), and T a biaxial tension reference stress. The parameter T, which should represent the biaxial yielding tension (al = 0 and 02 = a3 < 0), is the weakest point of the model. This value is to be considered only as a fitting parameter; it was the best available value for all the tests curves involved but does not correspond to the real tensile strength of ice. One has to keep in mind that the model presented 20
1 .
Reinicke and Reimer (1978) used this equation to analyse triaxial test data obtained by Jones (1978) and compared it to the other yield criteria mentioned above. Figure 2 reproduces the comparison they made. Since then, H~iusler (1983) has used the data he obtained from true (O 1 ::~ O"2 ¢ O'3)triaxial tests (H/iusler, 1981) to compare both Smith's and Pariseau's models. Figure 3 shows the result. YIELD CRITERION FORMULATION
account (MeUor, 1980). This function reproduces quite nicely the evolution of the deviatoric strength of ice as confining pressure increases. The ice exhibits some increase in strength up to 2 0 - 3 0 MPa and slowly decreases down to the phase change point. Using stress parameters defined in Fig. 5, the general expression of this function is written:
g
~ -20 E
FOR ICE, A NEW
As mentioned by Mellor (1980), these models were designed for rocks and soils and do not describe the feature which characterizes ice when subjected to hydrostatic pressure. Ice is known to change phases according to the phase diagram shown in Fig. 4. When dealing with pressures up to 200 MPa the liquidus decreases from O°C down to approximately -22°C. This particular feature of the ice and the data obtained by Jones (1982) from polycrystalline ice led the authors to propose a teardrop function as the best fit for these results (Nadreau, 1985). This teardrop shape satisfies the characteristics that a general failure criterion for ice should take into
"40
k•ll
\ \
I
0.2
\
V \
/I
\ \ \
I 0.4
0.6
0.8
Pressure , GPO
Fig. 4. Phase diagram for ice.
T
Fig. 5. "Teardrop" configuration.
C
79
rate is shown on Fig. 8. The evolution of the tail angle can be described by the following empirical relation:
here has been designed for multiaxial compression stress state only; since tension and compression fracture behaviours are completely different, one cannot expect a simple model to fit both fields. This model has been applied to the data presented by Jones (1982). The tests covered a wide range of strain rates from 1.4 X 10 -6 s-1 to 1.4 X 10 -2 s-a and were performed at different temperatures. The values considered as yield stresses are the maximum values encountered during the tests, which correspond, for low strain rates, to the initiation of plastic yielding and flow and, for higher strain rates, to brittle failure (Jones, 1982). The best fit curves for each strain rate series of Jones' results are presented in Fig. 6. It shows how the function covers the data perfectly. The tests were performed at -12°C. The hydrostatic pressure which makes the ice turn back to water at this temperature is approximately 130 MPa. The tension reference strength has been taken as 10 MPa for all the curves. Even if the differences between the theoretical curve and the experimental data points are magnified because of the expansion of the ordinate axis, the fitting appears quite good. When plotted in an isometric reference system the model looks quite appropriate for the multiaxial stress state (Fig. 7). It is worthwhile to note the evolution of the teardrop tail half angle with respect to strain rate. The plotting of the best fit values of 0 for each strain
0
=
K~ q
where q = 0.23 and K = 70.7 (0 being in degrees and 1 X 10 -s s -1 < ~ < 1 X 10 -2 s-X). The adequacy of this model to describe the confined compressive data leads the authors to use the teardrop shape to generate a three-dimensional yield surface for ice. In order to achieve the generalization, the isotropic behaviour of the granular ice is considered. This characteristic of the ice implies that the yield surface is symmetrical around the hydrostatic axis. The steps to generate the three dimensional surface are: first, generate the revolution surface around an axis (say, y) and then transform the equation in order to move the revolution axis from y to the hydrostatic axis. The starting equation being in the y, z plane (Fig. 9), Z
=
A t ( C - y ) O;
- -
T ) 1/2 ,
with A' = a/(C - 7') v2 and a = tan 0, the revolution surface equation can be written: x2+z 2 = A(C-y)2(y-T) withA =A '2. When using the transformation direction cosines
30
EL 0
G .
.
.
.
20
03
I0
--//A2,
°
,-.:l\\----
L.
.9 ° >
£3 0 -20
0
20
40
60
80
Confining pressure, 0-3,MPo Fig. 6. ]ones' triaxial tests.
I00
120
140
80
a 2 (x _y2) + (7(x + y ) + Gz) 2 - A (C-[3(x +y +z)) 2 X
shown in Table 1 and Fig. 10, (new coordinates being kept x, y , z in order to avoid prime superscripts), the equation becomes:
(~(x+y+z)-r)
= 0
140
120
I00
80
u Q,.
60
b--
4o
2o X
-20
- 20
0
2o
40
Confining
60
80
120
iO0
140
pressure, O'3,MPo
Fig. 7. Jones' triaxial tests. I00
7
I0
7
J i
=o o I
.
I 0 -e
.
.
.
.
.
.
i 0 -~
.
.
.
.
.
.
.
.
.
.
.
.
10.4 Strain
10.3 r a l e , s°I
Fig. 8. Evolution of the tail half-angle with respect to strain rate.
,
i
I
I I
i
10.2
I
I
i
I
I
I
il
i 0 -t
81
z
OZ
/
/
/
L._f j /
/
X
Fig. 9. Generating surface.
t
j ""
~
(7x Fig. 11. Yield surface for isotropic ice, new formulation. Fig. 10. Transformation of axes.
can be written as a function o f stress invariants; finally the model can be formulated as:
TABLE 1
f(oii)
Values of the angles and the direction cosines relating the new axes to the old set Angles x
y
z
54.7 54.7 54.7
114.1 114.1 35.3
Direction cosines x y z
= aI13 + bI12 + c i 1 + d J2 - 1 = 0
with a = 3a/TC 2 ,
b = -32(2 C + T)/TC ~ , x' y' z'
45 135 0
a --a 0
3 3 #
"r 6
If the development is carried out and the x , y , z are replaced b y a l , o~, o3 the expression o f the surface is:
A •3 (01 -k o"2 -[-0"3)3
A ~2 (2 C + T) (0"1 "t"0.2 "b0.3) 2 2 + A 3 C(2 T + C) (01 + 0.2 + 0"a) - § (02 - 0.3)2 -
-
+ (0.3-0.1) 2 + (0"1-0"2) 2 - A T C 2 = 0 This expression, being the yield surface o f a material,
c = 3(2T+C)/TC, d = -2/3A
TC 2 ,
I1 being.the first invariant o f the stress tensor and J2 the second invariant o f the deviatoric stress tensor. A representation o f this modelis presented in athree dimensional reference system in Fig. 11. It should be noted that the volume o f the teardrop defined in Fig. 11 increases as strain rate increases. Moreover, the temperature decrease makes the teardrop longer on the hydrostatic axis. These trends are valid when the temperature remains over -22°C and when strain rates range from 1 × 10 -6 s -1 to 1 × 10 -2 s -1. This model, developed for isotropic ice, could also
82 be extrapolated to describe the anisotropic behaviour of other types of ice. It would take the following form: f ( o i ] ) = al ( a y - Oz) 2 + a2 (tXz - ax) 2 + a3 (Ox - ay) 2 +
2 + a6 2 a4 J, y z + as Tzx Txy
+ a7 Ox + a8 o3, + a 9 0 z + (alo (~x + a l l ay + a12 a z ) 2
+ (a13 ax + a14 Oy + als Oz) 3 - 1 = 0 . For the complete anisotropic case fifteen coefficients would have to be determined from experimental tests. The number of coefficients would be reduced to ten if horizontal anisotropy is involved (as in $2 ice for instance).
CONCLUSION Using experimental data obtained from triaxial tests on polycrystalline ice, a model was developed which represents the behaviour of the ice when subjected to confining pressure. This model is an improvement over the existing yield criteria which failed in describing the effects of phase change due to hydrostatic pressure. The model presented here remains rather simple, and describes the ice behaviour more accurately at high confining pressures. It has been developed for isotropic ice but can be extrapolated to represent anisotropic types as well. Being based exclusively on ice data, it seems more appropriate than the other models derived from soils and rocks mechanics. It should contribute to the use of a proper yield criterion for ice in compression and should be useful to engineers.
ACKNOWLEDGEMENTS The information presented in this paper is part of a Ph.D. thesis undertaken under the supervision of Dr B. Michel at Laval University in Quebec City. This study has been possible thanks to a grant
from the Natural Sciences and Engineering Research Council of Canada.
REFERENCES Coon, M.D., Evans, R.J. and Gibson, B.H. (1984). Failure criteria for sea ice and loads resulting from crushing. Proc. IAHR 1984, Hamburg, Vol. 3, pp. 1-16. H/iusler, F.U. (1981). Multiaxial compressive strength tests on saline ice with brush-type loading platens. Proc. IAHR 1981, Quebec City, Vol. 2, pp. 526-539. H/iusler, F.U. (1983). Comparison between different yield functions for saline ice. Ann. Glaeiol., 4: 105-109. Jones, S.J. (1978). Triaxial testing of polycrystalline ice. Proc. Third Int. Conf. on Permafrost, Edmonton, Vol. 1, pp. 670-674. Jones, S.J. (1982). The confined compressive strength of polyerystalline ice. J. Glaciol., 20(98): 171-177. Mellor, M. (1980). Mechanical properties of polycrystalline ice. In Physics and Mechanics of Ice, IUTAM Symp., Copenhagen 1979, pp. 217-245. Michel, B. (1978). Ice Mechanics. Les presses de L'Universit6 Laval, Quebec City, 499 pp. Michel, B. (1981). Advances in ice mechanics. Proc. POAC 81, Quebec City, Quebec, pp. 189-204. Nadreau, J.P. (1985). Lois de comportement et de fluage de la glace granulaire simul~e de crates de pression. Ph.D. Thesis, Laval University, Quebec City, Quebec. Nadreau, J.P. and Michel, B. (1984). lee properties in relation to ice forces. Second state-of-the-art report, IAHR workhag group on ice forces. Proe. IAHR 1984, Hamburg, Vol. 4, Chap. 1. Pariseau, W.G. (1968). Plasticity theory for anisotropie rocks and soils. In K.E. Gray (ed.), Proc. Tenth Symp. on Rocks Mechanics, pp. 267-295. Ralston, T.D. (1977). Yield and plastic deformation in ice crushing failure. Proc. of the Arctic Ice dynamics joint experiment, Int. Commission on Snow and Ice Syrup., Univ. Washington Press, pp. 234-245. Ralston, T.D. (1978). An analysis of ice sheet indentation. Proc. IAHR 1978, Lulea, Sweden, Vol. 1, pp. 13-31. Reinicke, K.M. and Ralston, T.D. (1977). Plastic limit analysis with an anisotropic, parabolic yield function. Int. J. Rock Mech., Min. Sci. Geomech. Abstr., 14: 147-154. Reinicke, K.M. and Reimer, R. (1978). A procedure for the determination of ice forces - illustrated for polycrystalline ice. Proc. IAHR 1978, Lulea, Sweden, Vol. 1, pp. 217-238. Smith, M.B. (1974). A parabolic yield condition for anisotropic rocks and soils. Ph.D. thesis, Rice University, Houston.