anisotropic elasticity of polycrystalline ice ih

ELSEVIER Cold Regions Science and Technology, 22 ( 1994 ) 149-169

cold regions science and technology

Anisotropic elasticity of polycrystalline ice Ih

S. Nanthikesan, S. Shyam Sunder Massachusetts Institute of Technology, Department of Civil Engineering, Room 1-346, Cambridge, MA 02139, USA

(Received 15 February 1993; accepted after revision 29 May 1993

Abstract

A theory to determine the complete tensor of elastic moduli of generally anisotropic polycrystalline ice and its temperature dependence from the elastic properties of single ice crystals is presented in this paper. The model, expressed in closed-form, predicts the upper and lower bound limits of the elastic moduli for such polycrystals by generalizing the methods of Voigt ( 1910 ) and Reuss ( 1929 ), respectively, that were developed for isotropic aggregates. This involves obtaining the spatial average of the elastic moduli and compliances of individual crystals of ice by weighting them with the relative frequency of their orientations in the anisotropic fabric. Single ice crystals possess an open hexagonal structure and are transversely isotropic in their elastic properties. The theory is then applied to predict the elastic constants of transversely isotropic S1 and $2 ice, and orthotropic $3 ice. The predicted upper and lower bound limits are in excellent agreement with available experimental data.

I. Introduction

The ductile-to-brittle transition in ice is an im- portant phenomenon in engineering applications involving ice loads on structures and the bearing capacity of ice. In general, this transition is characterized by multiple modes of deforma- tion that include elastic and creep deformations, damage in the form of distributed cracking (viz., crushing) and extension of localized macro- cracks. The development of constitutive theories to characterize many of these complex deforma- tion processes requires the knowledge of the elastic and fracture properties of ice. This paper is concerned with estimating the elastic properties of polycrystalline ice.

Naturally formed ice in rivers, lakes or the sea is composed predominantly of long, columnar- shaped crystals of this type. Michel and Ram-

seier (1971 ) classified the fabric of these polycrystalline aggregates as a function of crystal or more specifically, c-axis alignment. The typical fabric can range from transversely-isotropic to more generally orthotropic. The mechanical properties of such ice are intrinsically anisotropic as well as temperature-sensitive. This is because in commonly encountered terrestrial ice (where, in general, homologous temperatures exceed 0.9 and pressures are less than 200 GPa), the constituent ice crystals possess a hexagonal polar structure [classified as ice Ih, (Hobbs, 1974)].

The influence of elastic anisotropy in charac- terizing crack nucleation and micro-damage processes associated with the ductile-to-brittle transition in polycrystalline ice has been studied by Cole (1988), Shyam Sunder and Nanthikesan (1990) and by Shyam Sunder and Wu (1990).

0165-232X/94/$07.00 1994 Elsevier Science B.V. All rights reserved SSDIO165-232X(93)EOO18-E

150 S. Nanthikesan, S. Shyam Sunder / Cold Regions Science and Technology 22 (1994) 149-169

These and other recent studies in ice mechanics have established the importance of elastic properties in the development of physically based constitutive theories applicable to the high end of the quasi-static regime of strain-rates (i.e., 10-4 to 10 -1 S - l ) .

The elastic properties of isotropic polycrystals of ice are available (for a review of the literature, see Gold, 1958 ) but corresponding information on polycrystals with fabric anisotropy is incom- plete. Theoretical estimates of the elastic constants for polycrystals may be derived from the elastic properties of the constituent monocrys- tals (see, e.g. Voigt, 1910; Reuss, 1929; Budi- ansky and Wu, 1962; Hill, 1965; Hashin and Shtrikman, 1962). In all these methods, however, the complete elastic tensor can be esti- mated only for a polycrystal with isotropic fabric.

Theoretical predictions for some of the elastic moduli for ice with an anisotropic fabric have been obtained in the past. As the global preferred orientation of c-axis (Fig. 1 a) strongly influences the mechanical properties of ice (Pounder, 1965), the type of ice determines its elasticity. Michel ( 1978 ) obtained the Young's modulus in the plane of isotropy of $2 ice; Sinha (1989) es- timated the two principal Young's moduli and the in-plane rigidity (shear) modulus for S1, $2 and a special case of $3 ice in which c-axes are distributed with a uniform probability density function in the preferred angular zone. The temperature dependence of these moduli were also determined. Both investigators used the averaging assumption of Reuss ( 1929 ) which theoretically yields only a lower bound estimate for the elastic moduli (Hill, 1952).

This paper presents a theoretical model to determine the complete elasticity tensor of generally anisotropic polycrystalline ice based on the elastic properties of the constituent ice single crystals. The model predicts the upper and lower limits of elastic moduli for such polycrystals by generalizing the methods of Voigt ( 1910) and Reuss ( 1929 ), respectively, that were developed for isotropic aggregates. This involves obtaining the spatial average of the elastic moduli and compliances of individual crystals weighted with the relative frequency of their orientations in the

C- AXIS X 3

X2

LANE F

X 1 (a)

X 3

x3 I

x 2

x 1

(b)

Fig. 1. (a) Single ice crystal. (b) Cartesian coordinate framework.

anisotropic fabric. The theory, expressed in closed-form, is then applied to predict the complete elasticity tensor for transversely isotropic S1 and $2 ice and orthotropic $3 ice with arbitrary distribution of c-axis orientations, includ- ing their temperature dependence.

S. Nanthikesan, S. Shyam Sunder / Cold Regions Science and Technology 22 (1994) 149-169 151

2. Background

This section briefly reviews: ( 1 ) the nature of the anisotropic fabric of polycrystalline S1, $2 and $3 ice; (ii) the independent elastic constants for S1, $2 and $3 ice; and (iii) the available experimental measurements of the elastic moduli for both monocrystalline and polycrystalline ice.

2.1. Fabric of polycrystalline ice

The fabric (grain size, shape and orientation) strongly influences the elasticity of ice (Poun- der, 1965 ). Michel and Ramseier ( 1971 ) classified widely occurring river and lake ice into sev- eral categories based on their fabric. This paper considers the three most commonly encountered ice types, viz., S1, $2 and $3 columnar-grained ice.

In S 1 ice, which is found in lakes, reservoirs, and rivers with low flow velocities, the preferred crystallographic orientation of the c-axis is vertical (Fig. 2a ), i.e., perpendicular to the ice cover. Grain sizes are large (in the order of centime- ters). In $2 ice, the preferred c-axis orientation is horizontal (i.e., in the plane of the ice cover) and randomly distributed with a uniform probability density function (Fig. 2b). The basal planes are parallel to the length direction of the columnar crystals. Both these categories of ice are transversely isotropic. In the presence of strong currents, while the c-axes may still be located in the horizontal plane, a strong preferred orientation can develop in the plane of the ice cover (Weeks and Gow, 1978). This type of ice is classified as $3 ice (Fig. 2c). It is usually found at the bottom of thick ice sheets of perennially frozen lakes, in river ice at great depth and arctic sea ice.

2.2. The independent elastic constants of ice

Hooke's law of linear elasticity is expressed as follows:

cro = C,~lE~z ( la)

or

eij = S~sklakt ( 1 b )

where eo and a o are the strain and stress tensors respectively, Sbkt is the fourth order elastic compliance tensor and C~kt is the elastic moduli tensor with respect to an arbitrary coordinate system (Xl, x2, x3), and ij,k,l= 1,2,3. Repeated indices imply summation. In its most general form, Sbkl (or C'Ok l ) has 81 constants. For a generally anisotropic Green elastic material, Sbkz (or C,~kl) has 21 independent constants which re- duces to nine in the case of orthotropic materials, to five for transversely isotropic materials, and to two for isotropic materials (Nye, 1957). The Voigt notation for Hookean elasticity of a Green elastic material is:

~=Str (2)

tlr= CE

where the compliance matrix, S, and elastic moduli matrix, C, are 6 6 matrices; the strain vector e = [ E1 ~, e22, e33, e32, e3~, e~2 ] x and the stress vector tr= [trl~, a22, a33, a32, a3~, a12] x. The symbol T denotes the transpose operation.

For an orthotropic material, the elasticity matrix a, with aij denoting either compliance or moduli (stiffness), can be expressed as:

a-----

B

all a~2 a13 0 a22 a23 0

a33 0 a44

Sym.

m

0 0 0 0 0 0 0 0 a~5 0

a66

The coefficients a44 , a55, a66 are the shear constants corresponding to the (x2,x3), (x, ,x3 ) and (x~,x2) planes, respectively.

For transversely isotropic (S1 or $2) ice, when the plane of isotropy is (xl,x2), [all=a22; a13=a23; aaa=a55; a66=2(a l l - a12) for compliance, and a66 = (a l l - - a~2)/2 for moduli] the elasticity matrix is:


a- -

w

a~l a~2 a13 0 a~j al3 0

a33 0

a44 Sym.

m

0 0 0 0 0 0 0 0 a44 0

a66

It should be noted that the form of the elasticity matrix for a crystal with hexagonal symmetry such as ice, is identical to that for the case of transversely isotropic polycrystal.

2. 3. Measurements of the elastic constants of ice

The elastic properties of single ice crystals are well established. Among the most systematic studies are those of Dantl (1969) and Gammon

et al. (1983). The former used a pulse-echo method in conjunction with a double-pulse in- terference technique. The experiments were con- ducted at temperatures ranging from the melting point to - 140 C on cylindrical test samples of 26 mm height and 30 mm diameter. The latter work used the method of Brillouin spectroscopy on local homogeneous regions of ice samples of arbitrary shape (contained in a 1 cm l cm 3.2 cm cell) to determine the elastic moduli at a temperature of - 16 C. The results of these two studies are summarized in Appendix A. These experimental data imply a variation in the Young's modulus of up to 41% with crystallographic orientation (Fig. 3 ).

The experimental techniques for determining the moduli ofisotropic polycrystals of ice fall under three categories: (a) static methods, includ-

(a) (b)

"l

MEAN DIRECTION /

SCA'HER ANGLE_ _ _ ' ~ 2 4 o

J r x I (c)

Fig. 2. Fabric of columnar ice. (a) S1 ice; (b) $2 ice; (c) $3 ice.

c- AXIS


15.0

T = -16C

e~

o.o

. . . . D

-15.0 - - GAMMON et al, (1913) ,

-15 0 15

E l (GPa)

Fig. 3. Compar ison of E rE3 surfaces at - 16C corresponding to the data of Gammon et al. (1983) and Dantl (1969).

ing tensile, bending and torsion tests (e.g. Hess, 1940); (b) dynamic (resonance) methods in which the natural frequency of the characteristic mode of vibration of a small sample is observed (e.g. Ewing et al., 1934); and (c) sonic and ul- trasonic methods, where the velocity of the wave is determined experimentally to obtain the re- lated elastic constants (e.g. Northwood, 1947).

Sinha (1989) pointed out the difficulties in separating the pure elastic response from other concomitant viscoelastic responses in polycrystalline ice. Gold (1958) suggests that grain boundary sliding renders the static results unre- liable for ice. Beltzer (1989) showed that the presence of grain boundaries makes it difficult to isolate the pure elastic response to wave propagation in polycrystalline materials. In the case of ice, where grain sizes are large (of the order of millimeters), the problem is aggravated. Conse- quently, the reliability of the measured polycrystalline elastic moduli is difficult to assess. The reported "reliable" values for the Young's modulus ranges from 8.69 to 9.94 GPa, while the rigidity modulus ranges from 3.36 to 3.8 GPa.

Elastic properties in the horizontal plane of $2 ice, viz. Young's modulus and Poisson's ratio, have been obtained by Sinha (1978, 1988). However, experimental measurements on the re- maining elastic moduli of such anisotropic polycrystals is lacking.

3. Review of prior theories for polycrystal elasticity

The theoretical methods proposed to determine the elasticity tensor of polycrystals fall into two main categories: (a) bound solutions that estimate the upper and lower bounds for the elastic moduli; and (b) self-consistent schemes that give direct estimates of the elasticity tensor.

Voigt ( 1910) postulated that polycrystals comprised of anisotropic single crystals could be replaced by an equivalent homogeneous uniform body. The effective modul i of the isotropic aggregate are taker to be a spatial average of the moduli of all the constituent crystals. This approach makes the assumption that a uniform strain applied to the polycrystal induces the same uniform strain in all the constituent crystals. The result is, therefore, approximate since local equilibrium, e.g. across grain boundaries, is generally not satisfied.

Reuss (1929) applied a similar approach but assumed that the compliance tensor of the isotropic aggregate is the spatial average of the compliance tensor of the single crystals. This approach implies that when a uniform stress is applied to the aggregate, all the crystals experi- ence the same uniform stress level. Reuss's method is also an approximation, since compatibility conditions, e.g. across grain boundaries, are generally violated.

Hill ( 1952 ), using Maxwell's reciprocity relations, showed that these approximate estimates (referred to as the V-R bounds in this paper) are, in fact, the upper and lower bounds, respectively, of the aggregate elastic moduli. In general terms, Hill's result can be stated as follows:

S0V~ ~ S,jkt ~< S~j~I C o~t ~


that form an isotropic aggregate. These are given below:

(8S , ' +3S3 ~ 2 4 - ' -- -~ ~S44 -Jr- ~Sl 3)


dictions on rigidity moduli and Poisson's ratios are made. For instance, Sinha ( 1989 ) notes that his rigidity modulus predictions for S1 ice results in an unrealistic value for Poisson's ratio that is greater than 0.5.

The closeness of the upper and lower bound estimates for isotropic ice suggest that for a mildly anisotropic ice aggregate, the V-R type bounds may be as accurate as the more involved self-consistent schemes or the variational approach. In the following section, the V-R theory is generalized to estimate the complete elastic tensor for an anisotropic aggregate. The generalized theory is then applied to predict the elastic constants of S 1, $2 and $3 ice.

4. Generalized theory for anisotropic polycrystals

Consider a polycrystalline aggregate that is replaced by an equivalent homogeneous body. The assumption contained in Voigt's method is applied first, i.e., under a given applied uniform stress tr all the grains undergo the same uniform strain ~ as the equivalent homogeneous (and anisotropic) body. The stresses in the aggregate are given by:

~r=Cc (5)

where C is the polycrystalline elastic moduli. The local stresses within each of the grains are given by:

tr' =C ' ( (6)

where C' is the compliance matrix of the single crystal with respect to the global coordinate system. The average stresses in all the grain_s must be in equilibrium with the applied stress or. Tak- ing the spatial average of Eq. (6) and noting the strain in every grain ~' is equal to (Voigt assumption ), we have:

~= ~ (7)

The symbol ( ) denotes a spatial average. Com- parison ofEqs. ( 5 ) and (7) yields the well known results of Voigt ( 1910):

C=(C ' ) (8a)

At the level of individual grains, the Voigt assumption allows adjacent grains to exist at different (uniform) stress states thus violating local equilibrium.

A similar analysis, with the Reuss (1929) assumption that all grains undergo the same uniform stress as the equivalent homogeneous body, yields the compliance matrix of the polycrystal which is expressed as:

S= (S ' ) (8b)

The Reuss assumption permits adjacent grains to have different deformations leading to local violation of compatibility.

The spatial average of the elastic properties of an anisotropic aggregate must be obtained by weighting the elastic constants at each crystal orientation by the relative frequency of that orientation. Recognizing that the elastic constants may be temperature dependent and using Voigt's assumption, this formulation yields:

~'( T) = ( C' ( O,O,T)F( O,O )dV (9a) V

and a similar expression results for the Reuss assumption:

~S( T) = f S' ( O,,T)F( O,O )dV (9b) V

where V is the representative volume considered; C' ( O,~,T), S' ( O,~,T) are the compliance and elastic moduli matrix, respectively, of a crystal with c-axis oriented at (0,q~) in the spher- ical coordinate system shown in Fig. lb; Tis the temperature; F(O,O) is the probability density function for the crystal (c-axis) orientation. This function is continuous and smooth and satisfies the following condition:

~F(O,O)dV=I (lO) V

The crystal elastic compliances and moduli at a given orientation (0,0) can be determined by the following coordinate transformation of the respective matrices Sg and Cg in the principal directions of the crystal:

C' ( T)=R TCg( T)R ( l la)

156 X Nanthikesan, S. Shyam Sunder/ColdRegions Science and Technology 22 (1994) 149-169

S' ( T) =RTSg( T)R (1 lb)

where R is the rotation matrix based on the direction cosines of the c-axis and R T is the transpose of the rotation matrix (see Shyam Sunder and Nanthikesan, 1990 and Shyam Sunder and Wu, 1990 for derivation). Each component of S' (or C' ) can be obtained by performing the nec- essary matrix multiplications on the right hand side of Eqs. ( 1 la,b) (see Appendix B). For instance, the Young's moduli of the monocrystal, E~, E~, for an arbitrary orientation making an angle ~ with the vertical c-axis can be obtained (see Appendix B for details) as:

E'I = S'IT i

= [ sin 4 ~Sl 1 + cos 4 ~$33 + cos 2 ~sin 2 ~ ($44 + 2S1 s ) ] - l

(llc)

E~ = S~5 I

= [cos 4 ~$11+ sin 4 OSss+ cos 2 Osin 2 0 ($44+2S~s) ] - l

( l id )

Equation (1 l c) is identical to the expression given in Fletcher (1970) for the variation of E't. The same transformation in Eqs. ( 11 a,b) gives the temperature dependence of the elastic constants in the current coordinate framework given the corresponding temperature dependence of single crystal elasticity in the principal directions.

4.1. Elastic constants of columnar S1, $2, $3 ice

The theory presented in Eqs. (9a,b) shows that the distribution of crystallographic orientation is the single factor which distinguishes the elasticity of S1, $2 and $3 ice, all of which contain identical single crystals. The most general form of this distribution occurs for $3 ice, where the c-axes may be non-uniformly distributed within a restricted region in the horizontal plane. Therefore, the theory is applied to this case first and the elastic constants under simpler situa- tions (e.g.: $2 ice) are then derived by special- izing this result.

$3 Ice: Case L The distribution of c-axis orientations for $3 ice can vary from site to site. There is evidence to show that it depends on the

velocity and direction of the current (Weeks and Gow, 1978 ). In general, this distribution is con- fined to a range of angles I 0o- ~Uo I ~< 0~< I 0o + ~uol, where 0o is the orientation of the mean c-axis and ~Uo is the scatter angle (Fig. 2c). The actual distribution function awaits experimental determination. Idealizations for this function should, however, recognize: (a) that all the c-axes are distributed in a single (horizontal) plane so that the distribution is a function of 0 only; and (b) the absence of c-axes oriented outside the preferred angular zone.

An appropriate probability density function which satisfies these requirements is the symmetric Beta distribution. This function yields a bell-shape distribution within two finite limits of 0, identically zero at and outside the limits, and is given as:

1 Fr~(o)=~mT~2m+l [0 - - (00 -~JO) ]m[O 0

+~0-0] m for 0o- ~,0 ~< 0~< 0o +~/o

F~'(0) =0 elsewhere (12)

where m is any integer which can be determined from the actual distribution if data are available and tim= [m!m!/(2m+ 1 )!]. A graphical repre- sentation of this distribution is shown in Fig. 4. Closed-form solutions for the elastic constants in Eqs. (9a,b) are not possible in this case. The compliances and moduli are obtained by numer- ically integrating the equations after substituting for F(O). A simple three-point Simpson's rule is

2.0 . . . . . 00 = 0 ~ 3 0

1.6 ~0 = 90

1.2 E rr~

0.8

0.0 ' ' ' -90 -72 -54 36 -18 0 18 36 54 72 90

0

Fig. 4. The probabi l i ty density funct ion for symmetr ic fl- distr ibution.


sufficient to integrate these equations accurately. $3 Ice: Case IL A simpler case in which the c-

axes are randomly oriented (with a uniform probability density function) in the horizontal plane between 0=01 and 0=02 (02 > 01 ) yields closed-form solutions for the elastic constants. This distribution was considered by Sinha (1989) and, as can be readily seen, is a special case of the distribution described by Eq. (12). F(0) now corresponds to a uniform distribution contained within the planar area and zero every- where else. This function F is normalized with respect to area and it follows from Eq. ( 10 ) that:

2 F(O)=02_O 1 for 01 ~ 0~ 02

=0 for 02~< 0~< 7~; --~z~0~01 (13)

Noting 01 = 0o- No and 02 = 0o + No and substituting in Eqs. (9a,b) yields:

0o+~o 1 rJ

J S'(0)d0 (14a) S= 2No 0o - ~o 0o +~uo

1 t~ I t C ' (0 )d0 (14b) C- 2No

~o - ~'0

The S' and C' matrices are obtained from single crystal elasticity as given in Eqs. ( 1 1 a,b). After lengthy but straightforward calculations and recognizing the hexagonal symmetry of the ice crystals, closed-form expressions for the orthotropic polycrystalline compliance matrix S are obtained as:

1 366 ~--'~0 [(No --4fl) (Sll +$33 -2513)

+ (No +4fl) 544 ]

NO--~ (511 "t-533-S44)

q- (3 No k- 2fl)Sl3]

1 313- 2No

1 $23- 2No

- - - [(No +oe)Sl2 + (No-a)S13]

__ __ [ (No_OL)Sl2-.]- (No-.I-o~)Sl3 ]

(15a)

and the elastic moduli matrix C is given by:

1 C~ -2No [bl Cll q- b2 C33 +2b3(C13 -]- 2C44) ]

1 C22 -2N 0 [b2 Cl i "Jrbl C33 +263(C13 + 2C44) ]

C33 ~--- Cll

1 c44 -2No [ (No + a) C44 + (No- c~) C66 ]

1 C55 -2No [ (No - ot~) C44 "~ (No "~- a) C66 ]

,[(1 ) C66--2~/0 4~0--] ~ (Cl l -1-C33-2C13)

-k (No -I'- 4fl) C44 ]

1 [blS1, +b2S33+b3(2S13+S44)] al l = 2-70-o

1 322 = ~ [b2S1, +b1533 +b3(2513 "~S44) ]

333 = 511

844_~. 1 [ ( ~/0 _~ o~ ) 844 .jl_ ( ~ffo _ o/) 566 ] 2No

355= l [ ( ~ff0 __ OL ) 544 ..~ ( ~//0 .jl_ OL ) S66 ] 2No

C12 ~--~0 [ ( lNo - ~) (C l 1 '~ C33- 4C44)

+(~No +2fl)C~3]

1 C13 = 2---~o [ (No +a)C12 + (No - 0~)C13 ]

1 C23 ~---~7- [ (No - 0~)C12 "~ (No ~- OL) C13 ]

Z~,o (15b)


where

3 3 b, =~,o +c~+/~, b~ =~,o-a+/~,

1 b3 =~o- fl

- 1 S13 =823 =2 (S12 "~ 813)

Sl2 1 =~ (SI1 -11-833 -$44 + 6S13)

$66 =2(S11 -S12) (17a)

and

1 1 4 a=~sin2~o cos20o, p=~sin4~,o cos 0o

$3 Ice: Case III. The classification of ice types by Michel and Ramseier (1971) suggests another distribution for $3 ice in which all the c- axes are more or less aligned along the direction of the current. This extreme situation corresponds to ~'o = 0 and, as to be expected, the Eqs. (15a,b) yield the elastic constants of the ice single crystal.

$2 Ice. The c-axes in $2 ice are uniformly distributed for all 0 ( -n~O~rt ) , and this defines the plane of isotropy. Using symmetry and con- sidering a quarter of a unit circle as defined by 0~0~


gle between the vertical and the axis of symmetry (Fig. lb). S1 ice. All the c-axes are aligned vertically in the case of S1 ice. Here, the transversely isotropic polycrystal and the constituent hexagonal single crystals have the same axes of symmetry. As a result, S 1 ice behaves like a pseudo-monocrystal, displaying the same elastic properties as those of the single crystal. Temperature dependence of elastic constants. The temperature dependence of the polycrystal elasticity is determined from the corresponding relationship for single ice crystals according to Eqs. (15a,b) and Eqs. (17a,b). Dantl (1969) measured the moduli of single crystals at a range of temperatures from -0 .7C to - 140C. The reported relations (Appendix A) were simplified by Gammon et al. (1983) who neglected the quadratic term without significant loss of accuracy. The resulting form is given by :

a(T) = a (Tm) ( 1 -qT) / ( 1 -qZm) (19)

where a denotes an arbitrary elastic constant with units of stress, T denotes the temperature in C, and Tm is the reference temperature at which the constant a is determined. The value of q was obtained from the Brillouin measurements to be 1.418 10- 3 deg- ~ (Gammon et al., 1983 ). The temperature dependence for all the elastic constants are taken to be identical and is given by Eq. (19).

5. Results and discussion

5.1. Choice of single crystal elasticity

The prediction of the polycrystal elastic constants depends on the accuracy of the measurements of the single crystal elastic constants. Their selection from the available monocrystalline elasticity data is, therefore, critical. The results of Dantl ( 1969 ) and Gammon et al. ( 1983 ), two of the more comprehensive experimental studies on the elastic moduli of the single ice crystals, are considered in this paper (see Appendix A).

In both these studies, the monocrystalline moduli are experimentally determined and the

corresponding compliances are computed from the known relationship between elastic compliances and moduli. The elastic constants presented by Gammon et al. (1983) satisfy the inverse relationship between the moduli matrix and the compliance matrix. However, the results of Dantl (1969) are found to violate this inverse relationship. For instance, at - 16 C, the compliances and the corresponding values obtained by inverting the measured moduli matrix differ by up to 41.7 % (Table 1 ). This discrepancy is explained by Gagnon (1993), who points out that Dantl ( 1969 ) recognized the large errors that result from inversion of his moduli matrix and used the compliance data of Bass et al. (1957) to determine the temperature dependence of compliances.

The compliances and moduli in Dantl ( 1969 ), therefore, cannot be used simultaneously to determine the upper and lower bound estimates of the elastic constants. The following single crystal elastic properties are considered: (i) the reported compliance data and moduli obtained from the inverse of this compliance matrix, and (ii) the reported moduli and compliances computed from the inverse of this moduli matrix. It is worth noting that Michel (1978) and Sinha ( 1989 ) used the compliances reported by Dantl ( 1969 ) in conjunction with the Reuss method to determine the polycrystalline elastic properties.

The compliances of single ice crystals at - 16 C for these three data sets are compared in Table 1 which shows that with the exception of S~3, the compliances reported by Dantl (1969) are within 2.5 % of those presented in Gammon et al. (1983). However, the compliance S13 in Dantl (1969) is smaller by as much as 24.8 %. The compliances computed by inverting the moduli matrix ofDantl (1969) are 2.7 to 11.9 % higher than the compliances of Gammon et al. (1983). The uncertainty of reported measurements of elastic constants in Gammon et al. (1983) are within 1.05 %, whereas in Dantl (1969) the maximum uncertainty for compliances is as high as 20 %. Independent experiments using Brillouin spectroscopy were con- ducted by Gagnon et al. ( 1988 ) to determine the pressure and temperature dependence of the

160 S. Nanthikesan, S. Shyam Sunder ~.Cold Regions Science and Technology 22 (1994) 149-169

Table 1 Comparison of the compliance of single ice crystal at - 16C (compliances in GPa t)

Gammon et al. Dantl ( 1969 ) (1983)

Reported Inverse of reported compliance moduli

Difference (%)

Sll 0.10318 0.10227 0.10959 $33 0.08441 0.08299 0.09040 $44 0.33179 0.32650 0.34619 $12 -0.04287 -0.04385 -0.04404 $13 -0.02316 -0.01856 -0.02630

7.2 8.9 6.0 0.3

41.7

elastic constants of single ice crystals. The measured values of elastic constants at atmospheric pressure are within 0.7 % of those reported by Gammon et al. ( 1983 ). In light of these facts, for subsequent analysis in this paper the single crystal elastic properties of Gammon et al. (1983) are used.

5 2. Validity of the bound estimates

S 1 ice is pseudo-monocrystalline. The same is true for the case of $3 ice when all the c-axes are parallel. The V-R bounds are, therefore, identical for these cases as seen from Eqs. (9a,b), ( 10 ) and ( 11 a,b) and hence, the predictions are exact.

The V-R bounds of the complete elasticity tensor of $2 ice are obtained from Eqs. (17a,b) for the monocrystalline elastic constants of Gammon et al. (1983). The limits are determined for the elastic moduli at - 16 C using Eqs. (17a,b), ( 18 ) and ( 19 ) and summarized in Ta- ble 2a. As is to be expected, the Voigt limits are consistently higher than the Reuss limits for all the constants but by not more than 4.2 %. The corresponding V-R bounds for the same elastic moduli based on the single crystal data of Dantl (1969) are obtained assuming the reported moduli to be reliable. The results are presented in Table 2b. The limits for the elastic moduli of $2 ice based on the moduli of Dantl (1969) are consistently lower than the corresponding values obtained using the data of Gammon et al. ( 1983 ) by not more than 6.3 %. The Voigt bounds are higher than the Reuss bounds but by not more than 3.2 % for the data ofDantl (1969).

Table 2 (a) Predicted Voigt and Reuss limits for the elastic constants of $2 ice at - 16 C (based on the elasticity data for single ice crystals determined by Gammon et al., 1983 )

Modulus Reuss Limit Voigt Limit Difference (GPa) (GPa) (%)

_EH 9.431 9.726 3.1 Ev 9.692 9.851 1.6 Gn 3.535 3.683 4.2 Gv 3.206 3.219 0.4

(b) Predicted Voigt and Reuss limits for the elastic constants of $2 ice at - 16C (based on the moduli data for single ice crystals determined by Dantl, 1969)

Modulus Reuss Limit Voigt Limit Difference (GPa) (GPa) (%)

EH 8.953 9.181 2.5 Ev 9.125 9.230 1.2 GH 3.340 3.451 3.3 Gv 3.061 3.072 0.4

The difference between the Voigt and Reuss bounds is comparable to the resolution of the limits, since the maximum variation observed in the experimental measurements of the monocrystalline elastic constants is about 1% in Gam- mon et al., (1983) and 7 % in Dantl (1969). Therefore, the upper or lower limit, or an average of these two can be taken as the actual elastic constants to an accuracy of 4.2 % when the single crystal elastic properties of Gammon et al. (1983) are used and 3.2 % when the moduli of Dantl (1969) are used. This validates the as-

S. Nanthikesan, S. Shyam Sunder / Cold Regions Science and Technology 22 (I 994) 149-169 161

sumption of Michel (1978) and Sinha (1989) who took the elastic constants of $2 ice to be the Reuss limit. The same assumption is adopted here as well.

der (1963), using seismic resonance experiments, found Poisson's ratio of sea ice to be independent of temperature in the range of - 3.6 to - 15C.

5.3. Comparison with experimental measurements

5.4. Comparison with previous theoretical predictions

Sinha (1978) measured the (horizontal) Young's modulus, EH, of $2 ice at -40C to be 9.5+0.3 GPa. The Poisson's ratio in the horizontal plane of $2 ice was measured at about 100 Hz and -20C (Sinha, 1988) and was found to lie in the range of 0.31 to 0.32. At -40C, the theoretical prediction for the Reuss limit of Young's modulus (based on the single crystal elastic constants of Gammon et al., 1983 ) is 9.74 GPa (see next section). The corresponding Voigt limit is 10.05 GPa. These limits overestimate the measured value by 2.5 % and 5.8 %, respectively. The Poisson's ratio in the horizontal plane is obtained from:

- 512 1 (20) Vn= Sll 2GH

The predicted values of Poisson's ratio using the assumption of Reuss and Voigt are 0.33 and 0.32, respectively. These values exceed the average experimental value of 0.315 by 4.5 % and 1.9 %, respectively (see Table 3 ). Eq. (20), in conjunction with Eq. (21 ) presented subsequently, predicts that this ratio is independent of temperature. While this has not been experimentally established for freshwater ice, evidence is available in the case of sea ice. Langleben and Poun-

Table 3 Comparison of the predictions of the elastic constants for $2 ice with experimental measurements (based on the single crystal data of Gammon et al., 1983 )

Experimental Theoretical Difference (%)

Er~ (GPa) 9.5+0.3 9.74-10.05 2.5-5.8 ( -40C) (Sinha, 1978)

V. 0.31-0.32 ( -20C) (Sinha, 1988)

0.32-0.33 1.9-4.5

The theoretical predictions for the Young's moduli of S 1 and $2 ice in the horizontal and vertical directions have been made by Michel (1978) and Sinha (1989) using the averaging procedure of Reuss (1929). Both predictions used the monocrystalline data of Dantl (1969). The theoretical model of Sinha (1989) differs from the Reuss's method in the following way: In the scheme of Reuss, first the polycrystalline compliance is obtained by averaging the single crystal compliances. Then, the polycrystal Young's (and rigidity) moduli is determined from the result, i.e., Eq. (18). Sinha (1989), however, computes the spatial averages of the Young's (and rigidity) moduli (not the compliances) of single crystals directly. One consequence of this approach is that numerical integration is required to obtain polycrystalline elastic constants. Secondly, the results are not strict lower bounds estimates for the elastic constants in the sense of Hill ( 1952 ).

The results of Michel (1978) and Sinha (1989) for S 1 and $2 ice are compared in Fig. 5 and Fig. 6, respectively, with the predictions of the theory presented in this paper based on Reuss's assumption and the single crystal elastic constants of Gammon et al. (1983). The small difference in the predictions for S 1 ice in Fig. 5 is due, in turn, to the difference in the results of Dantl (1969) and Gammon et al. (1983) which is within 1.8 %. The vertical Young's modulus of $2 ice is simply the single crystal compliance $33 (Eqs. 15a,b) and, hence, reflects the difference in the monocrystalline data used. The predicted horizontal Young's modulus is within 0.65 % of that of Michel (1978) and within 1.8 % of that of Sinha (1989).

Poisson's ratio in the plane of isotropy for S 1 ice is predicted to be 0.416 which is reasonable compared to the value of 0.51 obtained by Sinha


13.0

12.0

.2

~ 11.0

m

z 10.0

)-

(a) 9.0

S1 ICE

Ev = 11.58 [ 1 - 1.418 x 10 .3 (T-To) ] (GPa) ] F'H 9.48 [ 1 - 1.418 x 10 -3 (T-T0) ] (GPa)

- - MODEL (wKiammon et al., 1982 . . . . . MICHEL (1978) (wlDantl, 196~

~ " ~ ' ~ m w ~ ' - - SINHA (1989) (w/Dantl, 1969)

T = ~).2C o

-50 -40 -30 -20 -10 0

TEMPERATURE (C)

,.d

4.0

3.5

3.0

MODEL S l ICE

Gv

Gv = 2.95 [ 1 - 1.418 x 10 -3 if-To) ] (GPa)

GH = 3.35 [ 1 1.418 x 10 -3 if-To) ] (GPa) y =_0.2oC

2.5 , ~ ~ o

(b) -.so -40 -30 -20 -10 0 TEMPERATURE (C)

F ig . 5. Temperature dependence of the elastic constants ofS 1 ice. (a ) Young's modulus; (b) rigidity modulus.

(1989). This ratio for $2 ice is predicted to be 0.334 which is in close agreement with the experimental observations of 0.31-0.32 (Sinha, 1988); unlike the value of 0.393 predicted by Sinha ( 1989 ) which is significantly greater than the experimental value. This problem was recognized by Sinha ( 1989 ) who suggested that the approximate nature of the expression for VH ( = [EH/2GH ] -- 1 ) may be the source of discrepancy. However, this relationship can be shown to be theoretically exact (Appendix C). The problem is mainly due to the variation ascribed to the rigidity modulus GH with orientation. This variation, stated in Eq. (6) of Sinha ( 1989 ), is that of the torsional modulus which is determined from the rigidity moduli along planes resisting

(a)

10.5 $2 ICE - - MODEL (w/Gammon et al., 1983)

. . . . . MICHEL (1978) (w/Dantl, 1969) 10.2 m - - - - , SINHA (1989) (w/Dantl, 1969)

~~'"" ' " ' " E- V ,.... "~ .~

9.9 ~ ....... "~,.,..

9.6 '~ "~. . . . . . . .

- 3 ~G~ 9.3 ~ = 9.48 [ I - 1.418 x 10 .3 if-To) 1 ( /1 g.. = 9.2a [ 1 - 1.418 x 10 .3 i f -T ) 1 (GPa~ [ i . u / T - -02C '

9.0 ~ ~ ~ ~ ~- "

-50 ~10 -30 -20 -10 0 TEMPERATURE (~C)

4.0

3.5

3.0

S2 ICE

6v

V = 3.14 [1 - 1.418 x 10 "3 if-T0) ] (GPa)/

~n = 3.46 [ 1 - 1.418 x 10 -3 if-To) I (GPa t

2.5 ~ ~

{D) -50 -40 -30 -20 -10

TEMPERATURE ( 'C)

T =-0.2C o i

F ig . 6. Temperature dependence of the elastic constants of $2 ice. (a ) Young's modulus; (b) rigidity modulus.

the torque. Hence, it becomes a valid expression for rigidity modulus along a plane only when the material is isotropic. This can be readily seen by substituting the direction cosines into Eq. (6) of Sinha (1989). For instance, with q~=0, Go= 1/S44 is the rigidity modulus in the vertical plane. However, when 0=90 , G~# 1/S66 but (1/$44+ 1/$66)/2 which is the average of the rigidity moduli in the two planes. The result of this error is a systematic under prediction of the rigidity modulus GH leading to an over-prediction of P..

5.5. Temperature dependence of elastic constants

As mentioned earlier, the elastic properties of


the S I ice are those of the single crystals and 12.0 hence the elastic compliances of Gammon et al. (1983) with Eqs. (18) and (19) directly gives 11.0 the temperature dependence of the elastic constants of $1 ice to be: 10.0

m

Ev = 11.5811 -- 1.418X 10-3(T - - To) ]

EH =9.48[1-1.418x lO-3( T - To) ]

o~

9.0

$2 ICE

(T =-16C )

8.0 L i i i i J i i

(a) 0 1o 20 30 4o 50 60 7o 80 90 m Gv =2.95 [ 1 - 1.418X 10-3(T - To) ]

GH =3.35 [ 1-- 1.418X 10-3(T - To) ] (21) 3.8

where T, To are in C and To is the melting temperature ( -0 .2 C for freshwater ice). These re-

3.6 lationships are represented in Fig. 5. For $2 ice, this dependence corresponding to Reuss's assumption is obtained by substituting Eqs. (17a) ~ 3.4

and ( 19 ) into Eq. ( 18 ) (see Fig. 6 ): 3.2

' ' ' - ,~ ' ' S l ICE ' ' ' /

"~ (T = -16C) /

' , E l / /

i i i i i i t i

0 10 20 30 40 50 60 70 80 90

~, ANGLE FROM VERTICAL (~)

12.0

11.0

10.0

Z 9.0

O

3.8

S l ICE

( T = -16C )

G t G a

0 10 20 30 40 50 6O 70 S0 b 9o (~, ANGLE FROM VERTICAL

(a) 8.0

3.6

O 3.4

~ 3.2

3.0

(b)

Fig. 7. Anisotropy of the elastic constants of $1 ice at - 16 C as a function of the angle 0. (a) Young's modulus; (b) rigidity modulus.

~, ANGLE FROM VERTICAL

$2 ICE ( T = -16C )

~ 63

3.0 J i I i i i i i

(b} 0 10 20 30 40 50 60 70 80

~, ANGLE FROM VERTICAL (~)

90

Fig. 8. Anisotropy of the elastic constants of $2 ice at - 16 C as a function of the angle . (a) Young's modulus; (b) rigidity modulus.

m

Ev =9.4811 - 1.418)< 10-3(T - To) ]

EH =9.22[ 1 -- 1.418X 10-3(T - To) ]

Gv =3.14[ 1 - 1.418 x 10-3(T - To) ]

GH=3.46[1-1.418XlO-3(T-To)] (22)

The temperature dependence of the five elastic compliances obtained using the corresponding single crystal compliances of Dantl (1969) is given in Appendix D.

The elastic moduli ofS 1 and $2 ice show a mild dependence on temperature (Eqs. 21 and 22). The constants increase by 1.4 % per 10C drop in temperature for temperatures in the relevant range for most engineering applications (0 C to -50C) . The temperature dependence of the single crystal elastic constants of Gammon et al.


14.0

= 12.0 Jai-

l0 .0

z 8.0

O

6.0

(a)

~to = 40 S3 ICE

= 30 - - . ~ (T =-16C)

~o = 90~

UNIFORM DISTRIBUTION i i r i i i i i

10 20 30 40 50 60 70 80 90

ANGLE FROM MEAN C-AXIS DIRECTION (o)

6.0

K.3 5.0

,..l

4.0

3.0 (b)

$3 ICE cr = -16C

;' "Z \ :200 / \ k = 30

= 40 - -

= 90

0 10 20 30 40 50 60 70 80 90

ANGLE FROM MEAN C-AXIS DIRECTION (o)

F ig . 9. Anisotropy of the elastic constants of $3 ice with un i - fo rm distribution of c-axis orientations in the preferred angular zone as a function of scatter angle, ~Uo. (a ) Young's modulus; (b) rigidity modulus.

(1983), given in Eq. (19), assigns the same linear variation with temperature for all the constants. As a result, the polycrystalline moduli obtained from Eqs. (17a,b) display the same linear variation. Another consequence of Eq. (19) is that the predicted Poisson's ratio, #H, (see Eq. 20) is independent of temperature. As mentioned earlier, experimental measurements on sea ice by Langleben and Pounder ( 1963 ) show that the Poisson's ratio is independent of temperature in the range of -3 .6 to - 15C.

5.6. Magnitude of anisotropy in elastic constants

The anisotropy of S 1 ice is represented in Fig. 7 where the elastic moduli are plotted as a function of the orientation angle @ (the angle between the c-axis and the vertical global axis). The

12.0

11.o =

D 10.0

9.0

8.0

7.0 (a)

$3 ,CE m G 6oc

BETA DISTRIBUTION 1 i i i i i

0 10 20 30 40 50 60 70 80 90 ANGLE FROM MEAN C-AXIS DIRECTION ()

5.5

~" 5.0

I~= 4.5

4.0

3.5

3.0 ,v

2.5

(b)

s3 ,cE

m -oo / X (T =-I6C) = 30 = I0

=3

BETA DISTRIBUTION i i t i i i

10 20 30 40 50 60 70 80 90

ANGLE FROM MEAN C-AXIS DIRECTION ()

Fig. 10. Anisotropy of the elastic constants of $3 ice with symmetric fl-distribution of c-axis orientations in the preferred angular zone corresponding to @'0=90 as a function ofm. (a ) Young's modulus; (b) rigidity modulus.

Young's moduli vary by up to 41% with orientation. This maximum variation is higher than the difference of 23 % between the Young's moduli in the vertical and horizontal planes considered by Sinha (1989). The elastic constants of $2 ice are considerably less anisotropic (Fig. 8 ). For instance, the corresponding variation of the Young's moduli is about 10 %.

The anisotropy in the elastic constants of $3 ice depends on the distribution of c-axis orientations in the horizontal plane (see Fig. 2c for geometric details). The Young's modulus in the vertical direction is identical to that of $2 ice (Eqs. 17a,b) and, consequently, discussion is re-


stricted to the elastic anisotropy in the horizontal plane.

The elastic anisotropy is the largest when the c-axes are all parallel. The corresponding maximum variation in the Young's moduli is 41%. The anisotropy is reduced when the scatter angle ~u of the c-axis orientation increases (Fig. 9 ). The limiting case of this situation occurs when the scatter angle is 90 , i.e., uniform distribution of c-axes in the horizontal plane. The ice is then of $2 type and shows the least anisotropy for columnar ice.

For a non-uniform distribution of c-axis orientations, the elastic anisotropy in the horizontal plane increases. A symmetric Beta distribution (Eq. 12) for a scatter angle ~= 90 is considered in this analysis. By choosing the distribution parameter m, which determines the standard deviation of the distribution, the effective scatter angle can be varied. At m = 0, the distribution is uniform, corresponding to the lowest anisotropy level ($2 ice). At m = oo, the c-axes are all parallel (scatter angle ~,= 0 ) leading to the largest anisotropy (S 1 ice ). Fig. 10 shows the variation of one of the Young's moduli and a single rigidity modulus in the horizontal plane with orientation for different values of m. It can be seen that the elastic anisotropy is bounded by these two limits (m=0 and m=oo) and increases with increasing m.

6. Conclus ions

This paper has presented a theoretical model to predict the complete elasticity tensor of an anisotropic ice aggregate from that of its constituent crystals. The upper and lower limits of elastic moduli of the generally anisotropic aggregate is predicted by generalizing the earlier methods of Voigt (1910) and Reuss (1929), respectively, that were developed for isotropic polycrystals. The bounds are obtained by averaging the elastic moduli and compliances over a representative volume by weighting them with the relative frequency of their crystallographic orientations in the anisotropic fabric. The model is subsequently specialized for the case of three com-

monly occurring types of ice, namely, S 1, $2, $3. The results are presented as closed-form func-

tions of single crystal elasticity for S 1 and $2 ice and for $3 ice with a uniform distribution of crystallographic orientations in a preferred angular zone. For $3 ice with a non-uniform distribution of c-axis orientations, the solutions are obtained through a simple numerical integration scheme. In-situ experimental measurements must be obtained to estimate the actual probability distribution function for the c-axis orientations.

For elastic properties of single ice crystals, the studies of Dantl (1969) and Gammon et al. (1983) are considered in this paper. The elastic constants presented by Dantl (1969) violate the inverse relationship between the compliance matrix and the moduli matrix. The principal compliances of ice crystals reported in Dantl (1969) are within 2.5 % of the corresponding compliances determined by Gammon et al. (1983), with the exception ofS13 which is 24.8 % smaller. On the other hand, the compliances determined by inverting the moduli matrix of Dantl (1969), are higher by a maximum of 11.9 % than those reported by Gammon et al. (1983).

The V-R bounds for columnar ice are shown to give either exact results (for S1 ice and $3 ice with c-axes all parallel) or provide limits which are close enough to be comparable to the resolution of the available experimental measurements (for $2 ice). Therefore, either one of the bounds or an average of these bounds can be taken to give the approximate solution. The predictions are in excellent agreement with the limited reported experimental measurements for $2 ice (Sinha, 1978, 1988) as well as with previous theoretical approximations of Michel ( 1978 ).

Of the three types of ice considered, S 1 ice dis- plays the most anisotropy. The elastic constants vary by as much as 41% with orientation. $2 ice shows the least anisotropy with a maximum of 10.3 % variation in the elastic moduli with orientation. The anisotropy of $3 ice is determined by the scatter angle and the distribution of crystallographic orientations but is bound by the solutions for S 1 and $2 ice. With small scatter angle and parallel c-axes orientation the behavior is similar to S 1 ice. When the scatter angle is large


and the c-axes are uniformly distributed the behavior approximates $2 ice.

The elasticity of the ice polycrystals in this paper show very mild temperature dependence in the temperature range of 0C to -50C. All constants show the same linear dependence on temperature with a 1.4 % decrease in value for every 10 C drop in temperature.

In summary, the three main contributions of this paper are that: (1) Consistent and closed form expressions are derived to predict the complete tensor of elastic constants for the most commonly encountered types of polycrystalline ice with anisotropic fabric; (2) the proposed theory predicts values of Poisson's ratios that are physically correct and in agreement with the experimental data; and (3) the paper resolves the anomaly posed by the discrepancy between the experimentally determined values of Poisson's ratio and that predicted by the theory used in Sinha (1989).

Acknowledgments

The authors acknowledge the financial sup- port of the Office of Naval Research through its Sea Ice Mechanics Accelerated Research Initia- tive (Grant No. N00014-92-J-1208) and from an industry consortium consisting of Amoco, Arco, BP America, Chevron, Conoco, Exxon and Mo- bil through the MIT Center for Scientific Excel- lence in Offshore Engineering. The authors are indebted to R.E. Gagnon and R.L. Brown for their helpful comments.

Appendix A

Experimental results for the elastic constants of single ice crystals

Dantl (1969) Measured moduli (in GPa):

C~ ~ (T) = 12.904 [ 1 - 1.489 )< 10- 3T

- 1.85 10-6T 2 ] +0.3%

C33(T) = 14.075 [ 1 - 1.629)< 10-3T

-2.93)< 10-6T 2 ] +0.4%

C44(T) =2.819[ 1 - 1.601 10-3T

-3 .62 10-6T 2 ] +0.7%

C~2(T) =6.487[ 1 -2 .072 10-3T

-3 .62 10-6T 2 ] ___2%

G3(T)=5.622[1 - 1.874 10-3T] +7%

Reported compliances (in GPa - x ):

S~ (T) = 10.40)< 10-2[ 1 + 1.070

)< 10-3T+ 1.87 10-6T 2 ] -I- 1%

$33 (T) =8.48 10-2[ 1 + 1.405)< 10-3T

+4.66)< 10-6T 2 ] + 1%

$44(T) =33.42 10-z[ 1 + 1.505

)< 10-3T+4.04)< 10-6T 2 ] _ 1%

SI2(T) = --4.42 10-2[ 1 +0.463

10-3T-2.06)< 10-6T 2 ] +6%

St3(T)=- - 1.89 10-2[1+ 1.209

10-3T+6.15 10-6T 2 ] +__20%

where T is the temperature in C.

Gammon et al. (1983) (at - 16C) Measured moduli (GPa):

Cll = 13.929+0.41%

C33 = 15.010+0.46%

C44 =3.014+0.11%

C12 = 7.082 + 0.39%

C13 = 5.765 + 0.23%

Reported compliances (in GPa - ~ ):

S~ = 10.318 10-2+0.50%

833 =8.441 )< 10-2+0.45%

$44 =33.179 10-2+0.06%

Sl2 = -4.287)< 10-2+ 1.05%

S. Nanthikesan, S. Shyam Sunder / CoM Regions Science and Technology 22 (1994) 149-169 167

S13 : -2 .3168 10-2+_0.73%

Appendix B

Elastic compliances of single ice crystals with respect to an arbitrary coordinate f rame

8'~, = ( 1 -!~)28,~ +14833

+132(I -12) [28~3 +344]

833 ( 1 , 2 +n4833 t = --n.~) a l l

+n~(1 -n 2 ) [28~3 -Jl- S44 ] t 2 2 $44 =2 ( 1 -m~ -n 2 -2m2n~)S l ~ + 4m 3n 3833

+ 2( l _m2_nZ)S~z - 2 2 8m3n3S13

+ [n2(1 --m32) +m32( 1 --m~) +2m2n2 ]$44

8'12 2 2 "Jl- 833 -- S44 ) (1 2 2 =13m3($11 -- --13 - -m3)S l2

-- (l~ +m3 -212m2)S~3

S'13 2 z =/3n3(S11 "~ 833 --844 ) -- ( 1 --15 - -n2)812

-- ( l~ + n 2 -212n2)S13 (B.I)

Corresponding elastic stiffness constants

C'~ = (1 -/2)2C11 +14C33

+212(1 -12) [C,3 +2C44 ]

C33 ._~ ( 1 2 2 +n4C33 t _n3) CI 1

+2n2(1 -n 2 ) [C13 ll- 2C44 ]

C~4=2(1-m 2 -n2 - zm2nz)C~ t ~

+4mZn]C33 +2(1 -m2 -n~)C,2 2 2 -8m3n3C13 + [nZ(1 -m 2)

+m2(1 -m 2) - 2m]n~ ] C44 , 2 2 C12 =13m3(C~ ~ +C33 - 46"44) - ( 1 - l 2 -

m2)C,2

(12 2 2 2 -- +m 3 -213m3)C13

2 2 .~.. C33 _ 4C44 ) (1_12_n2)C12 C'13=13n3(C11

- (l 2 +n~ -21~n2)C~3 (a2)

Directions cosines

XI Z2 Z3

ZI ~2 ~3

Ii 12 13 ml m2 m3

nl n2 n3

(B3)

Direction cosines for crystal and global coordinate frames and 0 as defined in Fig. 2a

(Rotation 0 about x2 axis)

X, X2X 3

x~ cos0 0 -s in0 (B4) X2 0 1 0 X3 sin0 0 cos0

I I=COS~; 12 =0; /3 =- sin0; ml=0; mE----l; m3 = 0; n I = sin0; n 2 = 0; n 3 = COS~. Substituting in Eq. (B1),

E'l = S'll - l= [sin 4 ~811 +cos 4 q~$33

+cos 2 0sin z 0(844+2813) ] -~ (B5)

E~ = S~3 -1= [COS 4 0Sll+sin 4 0833

+cos 2 0sin 2 ~(844+2Si3) ] -1 (B6)

Direction cosines for crystal and global coordinate frames and 0 as defined in Fig. 2a

(Rotation about x3 axis )

3 1 ~(2 Z3 Z~ 0 COS0 sinO Z2 0 sin0 cosO

1 0 0

(B7)

11=0; /2=COS0; /3 =- sin0; m~=0; m2=s in0 ; m3----- cos0;, n I = 1; n2=0 , n3=0.


Determination of elastic compliances of $2 ice (Reuss's method) -- a sample calculation

From Eq. (13a), z~/2

S,1 =-2 f S',,dO (B8) 7~ 0

Substituting for direction cosines in Eq. (B7), S'11 becomes:

al l =COs40SI 1 +sin40S33

+ cos20sin20[2Sl3 +$44] (B9)

Substituting Eq. (B9) in Eq. (B8) and integrating,

1 S'~ =~ (3S11 -I- 3S33 3L S44 -k- 2S13) (B10)

Similarly, expressions for the other four compliance constants can be determined.

Appendix C

Consider transversely isotropic SI and $2 ice with the axis of symmetry (x3 axis) along the vertical direction. The elastic moduli can be expressed in terms of the compliances as (Sih and Chen, 1981):

~7H =El l ( =E22 ) = 1/$11 (C1)

VH = V12 = -S12/Sll (C2)

GH =G12 = 1/$66 = 1/2 (S~, -$12) (C3)

Substituting Eqs. (e l ) and (C2) in Eq. (C3)

G. =G,2 =bS11/2 ( 1 "q-~12 )

=bTH/2 ( 1 +~H) (C4)

Eq. (20) can be obtained by rearranging Eq. (C4).

Appendix D

Based on the compliances of single ice crystals determined by Gammon et al. (1983), the prin-

cipal compliances ture dependence ) and (19) ,as :

Sll =0.10844/(1

S33 =0.10522/(1

S44 =0.31899/(1

of $2 ice (and their tempera- are predicted from Eqs. (17a)

- 1.418X 10-3T)

- 1.418X 10-3T)

- 1.418X 10-3T)

S12 = -0.03619/( 1 - 1.418 X 10-3T)

$13 = -0.03377/( 1 - 1.418X 10-3T) (D1)

where T is the temperature in C. The compliances are in GPa- 1.

Similarly, based on the single crystal moduli data of Dantl (1969), the compliances of $2 ice in the principal directions are predicted from Eq. (17a). The results are approximated to the following form using regression analysis:

$1~ =0.11412/( 1 - 1.183 X 10-3T)

$33 =0.11148/( 1-0.981 x 10-3T)

$44 =0.33320/( 1 - 1.107X 10-3T)

$12 = -0.03896/( 1 - 1.283 X 10-3T)

S~3 = -0.03558/(1-0.691X 10-3T) (D2)

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