Viscous relations for the steady creep of polycrystalline ice

Cold Regions Science and Technology, 5 (1981) 141-150

141

Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

VISCOUS RELATIONS FOR THE STEADY CREEP OF POLYCRYSTALLINE ICE G.D. Smith and L.W. Morland Schoo/ o f Mathematics and Physics, University o f East A nglia, Norwich (U. K.)

(Received April 20, 1981; accepted in revised form June 10, 1981)

ABSTRACT Various published data from constant-stress creep tests on ice, relating minimum strain-rate to applied stress at different temperatures, are presented and compared. A temperature dependent rate factor is constructed from the Mellor and Testa (1969a) uni-axial compression data at uni-axial stress 1 . 1 8 X 1 0 6 N m -2 over the temperature range 212.15 K - 2 7 3 . 1 5 K. This factor is used to normalise the different sets o f data at different temperatures to a common temperature for comparison, but normalised strain-rates at a fixed stress still vary by a factor o f 3. Furthermore, it is shown that no alternative single rate factor will adequatety correlate the data at two different temperatures. A least-squares method is used to express the strain-rate as an odd polynomial in the stress; distinct polynomials are found for different sets o f data. Good matches are generally obtained over a uni-axial stress range 0--106 N m -2 by three terms:first, third and fifth powers o f stress; but less satisfactory non-monotonic polynomials involving negative coefficients are obtained in most cases if the seventh power is also included. Expressing the stress as an odd polynomial in the strain-rate, however, is not satisfactory, which is a reflection o f the shape o f the response at higher strain-rates. Inverse sinh function expansions failed in general, but inverse tan function expansions give good agreement to some data.

1 INTRODUCTION The response of polycrystalline ice to applied constant stress and constant strain-rate, as well as depen0165-232X/81/0000-0000/$02.50

dence on temperature, is reviewed in detail by Mellor (1980). This response is viscoelastic. There are elastic (instantaneous) strains in response to applied stress, but these are commonly negligible in comparison with the subsequent creep which occurs in three phases: a primary decelerating creep, a secondary, approximately steady, minimum strain-rate, and a tertiary accelerating creep. The time scales are strongly stress and temperature dependent. As stress and temperature decrease, the time to reach the secondary phase increases and may not be attained in laboratory tests. At these levels it is thought (Mellor 1980) that the distinction between tertiary and secondary creep diminishes. Low order differential relations can be constructed (Morland 1979, Morland and Spring 1981) to describe such viscoelastic fluid response when the elastic strain is neglected. Here we focus on viscous relations between stress and strain-rate to describe the secondary (minimum strain-rate) creep and the long-time limit, supposed steady, of the tertiary creep. It is the final tertiary steady creep that is important in glacier flow, but Glen (1975) points out that the tertiary creep is a result of recrystallisation which allows possible variation of structure and response, while the secondary minimum strain-rate is a property of the initial ice. He notes also that anisotropy can result from the recrystaUisation. Since all simple fluid laws (which exclude liquid crystal models) are necessarily isotropic in all configurations as a consequence of material frame indifference, the viscous fluid relations cannot describe such anisotropy. We will examine published laboratory data for the secondary creep, recognising that minimum strain-rate is not always attained, to express the strain-rate as low order polynomials in the stress for the different data

©1981 Elsevier Scientific Publishing Company

142 sets, and construct a temperature dependent rate factor to relate strain-rates at different temperatures. While good matches to individual data sets can be obtained, it is shown clearly that significant differences between data sets remain even after normalising for temperature difference: Any conclusion of good agreement between different data must be attributed to presentation of data and curves on logarithmic scales; we show the strain-rate-stress data and curves on linear scales. An alternative approach to the measurement of long-time steady response is to use ice-shelf data (Thomas 1971, Thomas 1973, Holdsworth 1974). While the uniformity of geometry and boundary conditions for large sections of shelf allow accurate theoretical models, correlation of measured surface features with the ice properties requires the flow solution for an assumed form of viscous relation. The above interpretations all assume a conventional power law with exponent 3, and estimate a single constant parameter in the relation. In addition, the estimate is based on an isothermal flow analysis for some mean temperature. The large temperature variation through an ice shelf and significant variation of creep-rate with temperature imply that the flow pattern is the solution of a more complex coupled thermomechanical problem, starting with a given temperaturedependent rate factor. Further theoretical analysis of such flow problems is important for fuller use of ice-shelf data in the deduction and confirmation of ice properties.

2 VISCOUS FLUID LAWS

It is generally accepted that elastic dilatation or compression is much smaller than the shear strains which occur over the time scales of natural ice mass flow. Budd (1969) reports that compressibility is less than 4 X 10 -l° (N m-2) -1 for pressures up to 3 X 107 N m -2, implying a maximum compression of order 10 -3 at the base of a large ice-sheet. With the conventional incompressibility approximation, the viscous fluid response can be expressed as a relation between the strain-rate tensor D (i.e. the symmetric part of the spatial velocity gradient) and the deviatoric stress S = a - ~-tr al, where a is the Cauchy stress. By material frame indifference, D is an iso-

tropic tensor function of S and S is an isotropic tensor function of D, which are expressed by the alternative Rivlin-Ericksen representations (Truesdell 1966) D = ~ k l ( J 2 , J 3 ) S + ~b2(J2,J3)[S 2 - ~2S 2 / ] ,

(1)

-, S = c k l ( I 2 , 1 3 ) D + d p 2 ( 1 2 , I 3 ) t[/9 2 -'~212Sl,

(2)

where the stress deviator and strain-rate invariants are defined by J1 = t r S = 0,

1 $2 , J2 = ~tr

11 = t r D = 0,

12 = ]tr/3a,

1

J3 = d e t S ,

(3)

I3 = d e t D .

(4)

Other sets of invariants may be chosen, but there are precisely tWO i---ndependent stress deviator invariants and two independent strain-rate invariants. These representations assume also that the response coefficients ~t,~2,¢1 ,~2 are independent of the mean 1 pressure p = --~tr a, which is reported to be a good approximation (Budd 1969, Glen 1975, Hooke et al. 1980) provided that the additional temperature dependence is expressed relative to the melting temperature for the given pressure. Temperature dependence will be discussed in the next section. The relation (1) with a different invariant J3 was deduced by Glen (1958) who noted that the determination of two functions ff 1, ~2, each with two arguments J2, J3, would require comprehensive tests varying and measuring two independent stresses and two independent strain-rates. It has been shown recently that a bi-axial stress configuration with two equal principal stresses is no more general than the common uniaxial stress configuration (Morland 1979), since both yield only.one independent deviatoric stress and one independent stress invariant. The necessary multiaxial stress tests to determine the general shape and dependence of the relation (1) have still to be performed (Hooke et al. 1980). To determine a viscous relation from uni-axial stress tests measuring one stress and one strain-rate, the simplification of (1) proposed by Nye (1953) has always been adopted, namely D = ~(J2)S.

(5)

The corresponding inverse relation is S = ~(I2)D,

(6)

143

where

~(J~)~(/2)

=

12

1,

= J2

~2(J2).

(7)

This simplification neglects the dependence of D on both the invariant J3 and the tensor S 2, and the dependence of S on Is and D 2. In particular, the assumptions ~2 = 0 and ¢2 = 0, imply that normal strain-rates are zero under simple shear stress, and normal stresses are zero in simple shear motion, neither of which has been confirmed to our knowledge. Glen (1958, 1975) notes that Steinemann's (1958) combined shear and compression data are not consistent with the form (5), and that dependence of ~k on a third invariant J3 may be necessary. Hooke et al. (1980) propose that experimental investigation of such dependence is required. The analysis of results must distinguish the effects of J3 and of a ~2 term. Dependence on an invariant J3 cannot, however, represent anisotropy. Budd (1969) also points out that the strain-rate at given stress decreases with increase of density, and uses extrapolated strain-ratedensity dependence found by Mellor and Smith (1967) to convert lower density data to a pure ice density 917 kg m -3. For simple shear stress r accompanied by a simple shear strain-rate ~, (5) and (6) become = ~(r2)r,

r = q~(~2)~,

(8)

since J2 = r~, /2 = q2. That is, the effective shear stress r e = ( J 2 ) 1/2 and effective shear strain-rate de = (/2) 1/~ defined by Nye (1953) reduce to the stress and strain-rate r and ~, in simple shear. The octahedral shear stress feet, on the plane equally inclined to the principal stress axes, and corresponding strain-rate doet, used by Budd (1969) to correlate different sets of data, are given by 2

roct = (~J2)

1/2

z2x1/2

= U~)

2

e o c t = (512)

1/2

re, /2-~ 1 / 2 *

= I.5)

ee •

(9)

In uni-axial stress o with corresponding axial strainrate d (and equal lateral strain-rates - 51 d by incompressibility), =

1

2

=

3 o5

3 3 "2 ° o = -~¢(~e )e.

2~tlo2)o,

(10)

We will use the relations (10) to construct the functions ~ (J2), ~b(12) from various uni-axial compressive stress data. Note that (10) implies symmetric responses in uni-axial compression and tension, and that non-symmetric responses would require dependence of ~k on J3 and/or if2 ~ 0. The invariance of the shear relation (8) under change of shear direction is preserved since J3 = 0 for simple shear stress and S 2 has no shear component. 3 TEMPERATURE DEPENDENCE

The conventional generalisation of the relation (5) to account for the significant temperature influence on creep rates is

D = a(T)tP(J2)S,

a(To) = 1,

(11)

where T is absolute temperature and the rate factor a(T) is normalised at some temperature To in the range of interest. We will adopt the melting point at zero pressure: To -- 273.15 K

(12)

That is, at constant stress, the strain-rate is proportional to a(T). It is found that the strain-rate increases significantly with T, more rapidly as the melting point Tm is approached, so

a ' ( T ) > O,

a"(T) > O,

T ¢ Tm.

(13)

The mean pressure effect noted earlier can be described by the transformation

T~T-Tm+To,

Tm= To-0.84X10-Tp

(14)

when p is measured in N m -2 (Budd 1969), and a(T) is the rate factor at p = 0. The MeUor and Testa (1969a) data at uni-axial stress 1.18× 106 N m -2 correspond to p = 3.9 × l0 s N m-2 and an insignificant decrease in melting point, so the associated a(T) can be treated as the factor f o r p = 0. Temperature dependence through a single rate factor in the form (11) is equivalent to a universal strain-rate-stress relation with rates measured with respect to a temperature dependent reduced time (Morland 1979) t [ a [T(t')] d t ' , fir=__ 0

0 -0t- =

0 a -afir -

(15)

144 This time scale transformation defines the thermorheologicaUy simple response of linear viscoelastic polymers (Schwarzl and Staverman 1952, Morland and Lee 1960), and has been proposed as a model of temperature dependence for the transient viscoelastic response of ice (Hutter 1975, Sinha 1978a,b, Morland 1979), though as yet without experimental confirmation. It is now well established (Mellor and Smith 1967, MeUor and Testa 1969a, Budd 1969, Glen 1975, Mellor 1980, Hooke et al. 1980) that the empirically based Arrhenius function

a(T) = A exp(-a/RT),

(16)

where Q is the activation energy and R is the gas constant (0.79 × 10 -2 kJ mo1-1 K-l), is not appropriate to temperatures above 263 K, which, o f course, is a range of considerable importance in natural ice masses. From lower temperature data, estimates of constant Q range from 50 k J / m o l - 8 0 kJ/mol (Budd 1969, Glen 1975, Barnes et al. 1971). For analytical purposes, a single smooth representation over the temperature range o f interest is useful, and we have used a least-squares fit o f the Mellor and Testa (1969a) data at uni-axial stress 1.18 X 106 N m -2 to the functions al(T) = 1 + a ( e aT - 1),

(17)

a2 (T) = ct 1et31T + ct2 e t32T ,

(18)

Note that the ratio al(T1)/al(T2) depends on both parameters a and/~, which is more general than the simple exponential form suggested by Budd (1969). While al(To) = 1 by (17), the form (18) allows a2(To) = aa + a2 :# 1 so that two coefficients are available in the least-squares correlation, and the data point T = To has no priority over other data points. At constant uni-axial stress, a(T) defined by (11) is simply the axial strain-rate ratio ~(T)/~(To). Table 1 shows the Mellor and Testa (1969a) data for polycrystalline ice of density 900 kg m -3 over the temperature range 212.15-273.15 K as the ratio e/co where eo = ~(To) = 4.06 × 10 -6 s -1. The values of al(T) and a2(T), constructed by a least-squares correlation, are also shown at the data temperatures for comparison, al(T) was constructed only over the temperature range 251.05-273.15 K, an important 20 K span, to provide a simple analytic estimate; the single exponential term is inadequate over the full data span. a2(T) was also constructed over 2 5 1 . 0 5 273.15 K, shown as a(~), and over the full range 212.15-273.15 K, shown as a~2). The data points and functions a 1,a~1), a~2) are shown in Fig. 1, with strainrates and a(T) represented by ln(~/~o) for convenience, but actual discrepancies are shown by Table 1. In view of the large ratio of strain-rates between T = 273.15 K and T = 212.15 K (or 251.05 K), equal priority to data points is achieved by minimising the sum

where ST = Z T = To+ 20T.

(19)

TABLE 1 Strain-rate-temperature data and approximate representations T(K)

T-

273.15 271.45 268.15 267.65 263.25 251.05 242.75 227.25 212.15

0 -0.085 -0.250 -0.275 -0.495 - 1.1:0~0 -1.5200 -2.2950 -3.0500

e/eo 1 0.595 0.2/28 A)?145 0.0706 0.0144 4.50 x 10 -a 4.41 X 10 -4 3.64 × 10 -s

a,

a~')

1.000

1.000

0.584 0.211 0.182 0.0546 0.0148

0.594 0.229 0.200 0.0700 0.0146

a~2) 1.068 0.530 0.201 0.180 0.0818 0.0132 3.88 × 10-a 3.95 X 10-4 4.26 X 10-s

or ST = ~ - - - ~

-1

(20)

145

I. (U~-) -3,5

-3.0

I

I

2,5 I

-2,0

-1 . 5

-1 , 0

-0.5

I

I

I

I

5 []

H&f

(1969A)

11.8=10

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

I 0 -I

!o

.//

i

.

I 0 -4

o:' t,o_ over the given set of temperatures, where a(T) is one of the forms (17) or (18). The parameters determined by both expressions are very similar, and give equally satisfactory correlation with the data. The Table and Figure show the results from the second form of ST. These results, with corresponding mean error E = ( S T / N ) 1/2 where N is the number o f data points, are E = 0.004,

a~l)(T) = 0.9316e 6.ss38~ + 0.0686e 1-441°~ , E = 6 X 1 0 -4,

(21)

a(2)(T) = 0.7242e 11.9567~ + 0.3438e 2-9494~ , E = 0.137. In contrast to the ratio 0.0144 at 251.05, Paterson (1969) suggests a ~ 0.1. Note that the Mellor and Testa (1969a) data correspond to a high effective shear stress 6.8 × l 0 s N m -2.

STRAIN-RATE-STRESS

if(J2) = 1.5k(3J2)

n-1 2

,

1 ~b(12) = -~k2- n (]I2)4

n-1 2n (23)

Fig. 1. Temperature dependent rate factor.

a l ( T ) = 1 + 0 . 9 8 6 ( e 6"~26T - 1),

(22)

where, from o = l0 s ~ 106 N m -2 at 273.13 K, Glen estimates k = 0.17 and n = 3.17 when o is measured in l0 s N m -2 and ~ i n a -1 = 3.18 X 10 -s s -1. Values of k given at lower temperatures are not consistent with the ratios e/Co given in Table 1. Glen notes that minimum strain-rates may not have been attained, and using Andrade's equation to extrapolate to minimum strain-rate deduces k = 0.017 and n = 4.2 at 273.13 K. The reliability o f Andrade's equation for predicting the minimum strain-rate has been refuted by Ting and Martin (1979) after detailed comparisons with experimental data. From the relations (10),

N/H ,

= ko n ,

RELATIONS

The longest serving axial relation is Glen's (1955) power law

which determines both ff and ~b explicitly. However, different values of the exponent n are estimated to correlate data o f different stress ranges, with n decreasing as stress level decreases (Glen 1975), so that no single power law (constant n) is satisfactory over a moderate stress range which extends to zero stress. Now ~b(I2) is a viscosity measure at effective strain-rate ee = I~/2, and for the power law (23) with n > 1, becomes singular as 12 -~0. Glen (1975) remarks that n does not necessarily reach unity as stress falls to zero, and that n > 1 is not theoretically ridiculous in this limit. The singularity is in fact apparent since the stress is given by the product CD which approaches zero as D ~ 0 for n > 1 , and a different factorization allows non-singular, but lengthier, analysis of the flow law and momentum equations (Morland and Johnson 1980). However, n > 1 in the limit is not confirmed by data, although Weertman (1973) argues on the basis of dislocation movement that low stress data reflecting n = 1 is necessarily transient and minimum strain-rate has not been attained. Bounded ~(I2) a s 12 -+0, corresponding to non-zero ~k(J2) as J2 ~ 0 , has more theoretical appeal. In fact, Lliboutry ( 1 9 6 9 ) h a s demonstrated an equally satisfactory match to the Glen (1955) data at 273.13 K by a three term polynomial in a, and Colbeck and Evans (1973) propose a similar polynomial representation, with different

146

coefficients, for their low stress data at melting point. Other functional forms for the axial relation, with finite viscosity at zero stress, have been proposed, and are summarised by Budd (1969) and Hughes (1972). We will present and compare various data, normalised to temperature 273.15 K by the function a(2)(T), and determine corresponding quadratic and cubic ~b(J2) by least-squares correlation. That is, polynomial relations

ee 6LEN (1955) 273.13 K GLEN (1955) 271.65 K x M&T (1969^) 273.15 K + STEIN (1956) 268.35 K

275

250

o D

Q

225

200 ×

175 ~50

ee

-

eO

Te

~ (r~) =

125

TO

×

•o[

(,o)' +c2( re}

-- Co+C, TO

,

-~o

+ ca

\Tol

iiOl

D

100

, (24)

\T'o I J

×

5 75

x ×

50

where, by (10),

× + ×

Te

=

( J 2 ) ~/2 =

(3)-1/2o,

ee = (/2) '/2 = }(3) 1/2~,

and eo = 200a -a,

re = 5 X 1 0 s N m -2

(26)

are chosen to normalise the ranges of strain-rate and stress covered. The various axial data are converted to the effective variables re, ee for presentation, and the coefficients relate to stress and strain-rate units l0 s N m -2 and a -1 respectively. Equal priority to the low stress data points is achieved by minimising the sum

Sr

= ~.

25

ro~e

over the data points (re,~e), to determine coefficients co,cl,c2,(c3 = 0), or Co,Cl,C2,C3, in the polynomials (24). Lliboutry's (1969) polynomial for Glen's (1955) data at 273.13 K for 0 < r e < 7.0 < l0 s N m-2, and Colbeck and Evans' (1973) polynomial for their own data at 273.15 K for 0 < re < 0.7 X 10SNm -= have the respective coefficients in the representation (24): L.:

Co=0.011 , e1=0.125, e2=0.756, c a = 0 .

C.&E.:

c0=0.0079, e1=0.394, c2=I1.61, c a = 0 . (28)

The polynomial constructions are not def'med, but we present the corresponding curves for comparison

++

x x

~x"" x

(25)

x

I

x x x ×

~

2

x

~>

~>.~ 3

4

+

+ 5

+ G

I 7

I 8

i 9

Te

Fig. 2. Various strain-rate-stress data. Strain-rate unit a - l , stress unit 105 N m-L

with data and our polynomial curves. Budd (1969) has presented graphically a variety of data converted to roct, eoct variables, mainly on logarithmic scales, and concludes that strain-rates at given stress are doubtful within a factor 2. We f'md much wider discrepancies between the different data sets. Figure 2 shows data from Glen (1955) at 273.13 K and a variety of temperatures with mean 271.65 K normalised to 271.65 K by a(2)(T), from Mellor and Testa (1969a) at 273.15 K and from Steinemann (1956) at 268.35 K. Clearly the Glen and Mellor and Testa data at the same temperature 273.15 K are incompatible. Figure 3 shows the same data normalised by a~2)(T), and in addition, two polynomial laws constructed by Butkovitch and Landauer (1958) from commercial ice and tunnel #1 ice at 268.15 K, here normalised by a~2)(T). There are still wide discrepancies between the different data sets after normalisation. The B. & L. commercial ice polynomial is not consistent with any of the other data, while the tunnel #1 polynomial lies between the MeUor and Testa and Steinemann data. Alternative Butkovitch and Landauer (1958) power law and sinh curves offer no better agreement. Figure 4 shows a 3 times magnification of the low stress region of

147

I

~e 275 258 225

275 [3

250 5]

/

/

°

/

108

,'

75

i./t //

25

+

100

3

Fig. 3. D a t a n o r m a l i s e d b y

!

,~N!

I

: /

i

f

L I

/,

!

//D

I

50

+

25

, re

+ 4

i

75

+

~

2

' / ,,//

~25

+

x

~ x

~,~zx".....*~

,

150

/

x x/ N

/+

,, %

,/

1

/ .

'/ .1

58

/ / ,-~

r~, / //,

;75

X

I

125

/

208

+

i

150

/ r~' .; .~ .

53

/ f

175

[] GLEN ( 1 9 5 5 ) 2 7 3 . 1 3 K - GLEN ( 1 9 5 5 ) POWER LAW . . . . . LLIBOUTR¥ ( 1 9 6 9 ) POL¥ . . . . . . S&M 3-TERM POL¥ . . . . S&M ,-TERM POLY

225

/

200

/

ee

/ I o GLEN (1955) 273.13 K / O GLEN (1955) 271.65 K / x I~.T (1969^) 273.15 K / + STEIN (1956) 268.35 K / --BII, L (1959) TUNNEL#1 ICE/ . . . . BI,L (1958) COMMERCIAL ICE /

5

7

G

a~2)(T)

8

[3 ~ . , ~ 1

£

2

I 3

I 't

I 5

I 6

t 7

,I

i

're

9

Fig. 5. G l e n 2 7 3 . 1 3 K normalised data and correlated f u n c t i o n s

and t w o B u t k o v i t c h

and Landauer p o l y n o m i a l laws. Strain-rate unit a-1, stress unit l 0 s N m -2.

o f stress. Strain-rate unit a - 1, stress unit 1 0 s N m -2.

Fig. 3, together with two Mellor and Testa (1969b) low stress data points and the Colbeck and Evans (1973) polynomial law (282). The wide difference between the Mellor and Testa (1969a) data and other data at effective shear stresses below 2.5 X l0 s N m -2 is clearly shown, whereas the various other data are in reasonably good agreement. Note that the C. & E. polynomial was constructed only for effective shear stress below 0.6 X l0 s N m -2.

Figure 5 shows the Glen (1955) data at 273.13 K together with the power law (22), Lliboutry's (1969) polynomial, and our 3- and 4-term polynomials (24) constructed by minirnising Sr in (27). The corresponding coefficients and mean error E = (S~.[N)1/2 are

3-term: co = 8.34X 10 -3, cx = 0.200, c2 = 0.463, E = 0.257.

(29)

4-term: co = 0.955 × 10 -2, Cx = 0.134, c2 = 0.906, c3 = --0.388, E = 0.251.

ee G 30 25 2e

x +

GLEN (1955) 273.15 GLEN (1955) 273.13 K GLEN (1955) 271,G5 K MST (1969A) 273.15 K M~T (19698) 271,09 K STEIN (1956) 268,35 K C~E (1973) 273,15 K

/ #

/ #

/ x

x

/

/

15. 18. 5_

×× ×N ××

i

x

×

x

×

x

N

~ ~

: ~ 0.5

~ - - ' ~ 5 " ~ " ~ "~ 1.8

~q"!

v 1.5

I 2.~

0 O~

6, Te

Fig. 4. L o w stress n o r m a l i s e d d a t a and t h e C o l b e c k and Evans p o l y n o m i a l law. Strain-rate unit a -1 , stress unit l 0 s N m -2 .

Correlation appears excellent for the low stress data points, as a result of the ratio form of St in (27), and the mean error E reflects the strain-rate discrepancy at fixed stress at the high stress data points. The curves show small stress discrepancy at fixed strainrate. The Lliboutry polynomial appears good at low and high stresses, but shows greater deyiation at intermediate stresses. Glen's power law is less satisfactory. The curve (292) is not monotonic, which is associated with the negative coefficient c3, and this feature is found for all the 4-term polynomials for the data sets shown in Fig. 3. They therefore define multivalued

148

Ce 275

2513

225

GLEN ( 1 9 5 5 ) 271.65 K x M&T ( 1 9 6 9 A ) 273.15 K + STEIN ( 1 9 5 8 ) 268.35 K -3-TERM POL¥ GLEN 2 7 3 , 1 3 K ........ 3-TERM POLT M&T 273,15 K ..... 3=TERM POL¥ STEIN 2 6 8 , 3 5 K .... 3-TERM POL¥ GLEN 2 7 1 . 6 5 K

i / i

2oo

:

+l

175

/ j

/

,

15o

¢_

I

l OO

/

x~ /

5o x -~'~"~

.

.

1

2

+

a~2)(T,)/a~2)(T~),

/ ,/

•

~-~

3

/

/

," i ," / /' / x:" / ,, / ," / ~ . x .,," x/ /.,4x ,'" / ...... .. / . ~ + / / ~,.. . I

75

25

/

/

:' ,;

125

/

~t

5

Te 6

7

8

(1969a) data. There are still wide discrepancies between the various data. We now ask if there is a single rate factor a(T) which makes the various data consistent with the relation (11); that is, if the ratio of strain-rates at each fixed stress from different constant temperature data is independent o f stress. Figure 7 shows strain-rate ratios between various constant temperature data, constructed from the corresponding curves (291) and (30) multiplied by the respective a~2)(T) to eliminate the above normalisation. The values in brackets for each curve represent the ratio where T1,T2 are the respective data temperatures. These strain-rate ratios are clearly not independent of stress, and their mean values are not close to the bracketed values implied by the Mellor and Testa normalisation. Thus no pair of the data sets considered is consistent with the existence of a single temperature rate factor a(T). We see that the ratio AID changes by a factor 3 while A[B and BID change by a factor 2 between effective shear stresses l0 s N m -2 and 8 X l0 s N m -~. However, the only extensive temperature data at fixed stress are the Mellor and Testa (1969a) data used to construct a~2)(T). We have also applied the least-squares correlation of the viscous data, normalised to 273.15 K by a~2)(T),to the inverse forms

£

Fig. 6. V a r i o u s n o r m a l i s e d d a t a a n d c o r r e l a t e d 3 - t e r m p o l y n o m i a l s in stress. Strain-rate u n i t a -~, stress u n i t l 0 s N m -2.

functions of strain-rate for the stress, and would not be satisfactory for use in this inverse sense. Figure 6 shows 3-term curves for the normalised Mellor and Testa (1969a), Steinemann (1956), Glen (1955) at 271.65 K, together with the curve (291) for Glen (1955) at 273.13 K for comparison. The corresponding coefficients and mean errors are

~e ratios 60:.

A B C D

50

M . & T . : Co = 0.311, cl = - 0 . 3 3 8 , c2 = 0.251,

= 6e (GLEN 2 7 3 . 1 5 K) = ee (GLEN 22"1 .65 K) = ~=e (M&T 273. 15 K) = ee (STEIN 2 6 8 . 3 5 K

)

~

E = 0.397. Stein.:

Co = 0.00264, cl = 0.0595, c2 = 0.0302, E = 0.312.

(30)

40

~

A/D

<5.03)

3O

Glen (271.65 K): Co = 0.00139, Cl = 0.0708, c2 = 0.0731, E = 0.316. In spite of the negative cl in M. & T., the curve is still monotonic, but has an inflexion point. The curves (30) are very different from the curve (291) obtained from Glen's data at 273.13 K. The 3-term laws (291) and (30), shown in Figs. 5 and 6, are constructed from respective data normalised to temperature 273.15 K by the rate factor a~2)(T), (213), given by the Mellor and Testa

2O

10 . ' . \ ,

A/B "~--

(1.851

............

BID

.................................. ~ .......... ', I

i- .... 2

I=-3

I 4

I 5

.-(2.71)

*/c - % . % ~ I 6

I ?

I 8

Te

Fig. 7. Strain-rate r a t i o s b e t w e e n v a r i o u s c o n s t a n t t e m p e r a ture data. N u m b e r s in b r a c k e t s are t h e c o n s t a n t r a t i o s d e f i n e d b'y a~2)(T). Stress u n i t l 0 s N m -2.

149

~e

're =

~(ee)

-

I"0

273.13 K data, with b0 = 0.2299, b~ = 371.6, b2 = 0.4957 and b3 = 4.7468. The minimisation calculation is sensitive to starting values chosen for the coefficients b0,b~ ,b2 ,b3, and these were chosen to be consistent at low stress with the satisfactory expansions (24) for the different data. No satisfactory inverse form (31) was obtained for the Mellor and Testa 273.15 K data because of the high strain-rates at low stress.

,2

=

e0

ae(bo+b,(f

lee\4

de 6

eo

or bosinh-i(b,~-l+b2sinh-'(b3~o), eo I

o,

bo tan_l(bl ee

ee

(31)

CONCLUDING REMARKS

The 3- and 4-term polynomials (31 ~) were not satisfactory for any of the data, giving a mean error E range 0.5-+0.75, essentially because of the decreasing curvature required as ee increases. In general the inverse shah function expansions (312) were poor, with mean error E up to 0.81, but gave excellent agreement, E = 0.071, for the Glen 273.13 K data with bo = 1978.5, b~ = 7.4 X l f f s, b2 = 0.1193 and b 3 = 791.1. Figure 8 shows the (311,2) curves for the Glen 273.13 K data. Figure 9 shows the inverse tan function expansion (313) for normalised Glen 273.13 K, Glen 271.65 K, and Steinemann 268.35 K data, with respective mean errors E = 0.049, 0.140, 0.173. Again there is good correlation with the Glen

It is clear that there is no real consistency between the various data, with temperature normalised strainrates at fixed stress varying by a factor 3. Presentation of data on log scales can conceal the variation. Echoing Hooke et al. (1980), there is need for more controlled laboratory tests, and in particular multiaxial stress tests to determine any influence of the third invariants Ja and I~, and of the tensors S 2 and D 2, in the viscous relation. It is also evident that present data do not predict a single temperature dependent rate factor. Further theoretical and experimental investigations of ice-shelf creep offer prospects for determining long-time steady response properties of polycrystalline ice.

I" e

•r e

9 +

+

8 + /

/'

,

/ ,.--.,

.,i .

'-,.,

./

[] []

\.\

',,,

,.' /

."

[]

,,"

',,

\,

./ / , K"

,

,~ []

/'

./

I

5

', ,,

4

:"//

[]

GLEN(1955)

--ARCSINH "/ .~/

....... 4-TERM .... 3-TERM

273.1~

+

/

'

~

3

K

[] GLEN(1955) 273.13 K <) GLEN ( 1 9 5 5 ) 2 7 ] . 6 5 K + STEIN ( 1 9 5 6 ) 2 6 8 . 3 5 K --ARCTAN TO GLEN 2 7 3 . 1 3 K . . . . . . ARCTAN TO GLEN 271 . 6 5 K .... ARCTAN TO S T ~ ] N 2 6 • . 3 5 K

TO GLEN 273,13 K POLY. POLY.

TO GLEN 2 7 3 1 1 3 TO GLEN 2 7 3 1 1 3

/ ,'/

K K

2

1

i

t

25

50

~

75

p

i

t

b

100

125

150

175

~

200

i

t

I

225

250

275

~e

I

I

r

P

I

I

I

I

I

I

25

50

75

]OO

125

150

175

200

225

250

Fig. 8. Glen 273.13 K normalised data and correlated functions o f strain-rate. Strain-rate unit a-1, stress unit 105 N m -2. Fig. 9. Various norrnalised data and correlated functions o f strain-rate. Strain-rate unit a -1 , stress unit l 0 s N m -2.

ee 275

150

ACKNOWLEDGEMENT This work was performed as part o f NERC projects on Interactions between Glaciers and Underlying Deformable Materials and on the Relationship between Glaciers and Climate during the last Glacial Period in N.W. Europe.

REFERENCES Barnes, P., Tabor, D. and Walker, J.C.F. (1971), The friction and creep of polycrystalline ice, Prec. R. Soc. A, 324: 127-155. Budd, W.F. (1969), The dynamics of ice masses, ANARE Sci. Rep. A (IV) Glaciology, No. 108, 1-216. Butkovitch, T.R. and Landauer, J.K. (1958), The flow law for ice, IASH 47, (Symposium Otamonix): 318-327. Colbeck, S.C. and Evans, R.J. (1973), A flow law for temperate glacier ice, J. Glaciology, 12:71-86. Glen, J.W. (1955), The creep of polycrystalline ice, Prec. R. Soc. A, 228: 519-538. Glen, J.W. (1958), The flow law of ice. A discussion of the assumptions made in glacier theory, their experimental foundations and consequences, IASH 47, (Symposium Chamonix): 171-183. Glen, J.W. (1975), The mechanics of ice, CRREL Monograph 11-C2b: 1-43. Holdsworth, G. (1974), Erebus glacier tongue, McMurdo Sound, Antarctica, J. Glaciology, 13: 27-35. Hooke, R. Le B., Mellor, M., Jones, S.J., Martin, R.T., Meier, M.F. and Weertman, J. (1980), Mechanical properties of polycrystalline ice: an assessment of current knowledge and priorities for research (ICSI Report), Cold Reg. Sci. Tech., 3: 263-275. Hughes, T. (1972), Thermal convection in polar ice sheets related to the various empirical flow laws of ice, Geophys. J.R. Astr. Soc., 27: 215-299. Hutter, K. (1975), A general non-linear viscoelastic plate theory and its applications to floating ice, Acta Mech., 21: 313-327. Lliboutry, L.A. (1969), The dynamics of temperate glaciers from the detailed viewpoint, J. Glaciology, 8: 185-205. Mellor, M. and Smith, J.H. (1967), Creep of snow and ice, in Physics of Snow and Ice, Conf. Sapporo 1966, Prec. 1: 843-855.

Mellor, M. and Testa, R. (1969a), Effect of temperature on the creep of ice, J. Glaciology, 8: 131-145. Mellor, M. and Testa, R. (1969b), Creep of ice under low stress, J. Glaciology, 8: 147-152. Mellor, M. (1980), Mechanical properties of polycrystalline ice, in Physics and Mechanics of Ice, IUTAM Symp., Copenhagen 1979, Prec., P. Tryde (Ed.), Springer Verlag, Berlin: 217-245. Morland, L.W. and Lee, E.H. (1960), Stress analysis for linear viscoelastic materials with temperature variation, Trans. Soc. Rheol., 4: 233-263. Morland, L.W. (1979), Constitutive laws for ice, Cold Reg. Sci. Tech., 1: 101-108. Morland, L.W. and Johnson, I.R. (1980), Steady motion of ice sheets, J. Glaciology, 25: 229-245. Morland, L.W. and Spring, U. (1981), A viscoelastic fluid relation for the deformation of ice, Cold Reg. Sci. Tech., 4: 255-268. Nye, J.F. (1953), The flow law of ice from measurements in glacier tunnels, laboratory experiments and the Jungfraufirn borehole experiments, Prec. R. See. A, 219: 477489. Paterson, W.S.B. (1969), The Physics of Glaciers, Pergamon, Oxford. Schwarzl, F. and Staverman, A.J. (1952), Time-temperature dependence of linear viscoelastic behaviour, J. Appl. Phys., 23: 838-843. Sinha, N.K. (1978a), Short-term theology of polycrystalline ice, J. Glaciology, 21: 457-473. Sinha, N.K. (1978b), Rheology of columnar-grained ice, Exptl. Mech., 18: 464-470. Steinemann, S. (1956), Flow and recrystallisation of ice, IASH 39, (Tome IV, Assemblee Generale de Rome 1954): 449-462. Steinemann, S. (1958), Experimentalle Untersuchungen zur Plastizit/it yon Eis. Beitr~ge zur Geologic der Schweiz, Geoteehnische Serie, Hydrologic, Nr. 10. Thomas, R.H. (1971), Flow law of Antarctic ice shelves, Nature, Physical Science, 232: 85-87. Thomas, R.H. (1973), The creep of ice shelves: interpretation of observed behaviour, J. Glaciology, 12: 55-70. Ting, J.M. and Martin, R.J. (1979), Application of the Andrade equation to creep data for ice and frozen soil, Cold Reg. Sci. Tech., 1: 29-36. Truesdell, C. (1966), The Elements of Continuum Mechanics, Springer Verlag, New York. Weertman, J. (1973), Creep of ice, in Physics and Chemistry of Ice, E. Whalley, S.J. Jones and L.W. Gold (Eds.), Royal Society of Canada, Ottawa: 320-337.