Specimen preparation

In order to observe the rotation process of the C-axis of individual grains during the compressive deformation, thin specimens 2 mm thick and 30 x 20 mm2 in area were prepared from both Dye-3 ice core and artificially grown polycrystalline ice. The latter was grown from the surface of pure water at a moderate rate, so that the c-axes became horizontal under a certain depth. When thin specimens were cut horizontally from this ice, it had two-dimensional random orientation fabric of c-axes. Its average grain size was 2 mm.

In the case of Dye-3 ice core (sample 1851, 1900 m depth and sample 1950, 2000 m depth), thin specimens were cut from the core so that the plane of the wide area was parallel to the core axis and its long axis made 45° with the core axis. Since both cores have the single maximum fabric, the c-axis of each grain lies on the plane of the thin specimen and angles between the compressive axis and the mean direction of c-axes of both cores are 40° and 60° respectively, with a standard error of 10°. Average grain sizes for the cores of sample 1851 and sample 1950 were 1.5 mm and 2.7 mm respectively.

For a study of the effect of compressive deformation on the random fabric, normal compressive tests were carried out with bulk specimens 20 x 20 x 70 mm². They were cut from Dye-3 ice core no. 120 (201 m depth) so that the long axis of the specimen was parallel to the core axis, and therefore, the specimen’s compressive axis in the test coincided with that of the natural compressive stress in its original depth in the ice sheet. The c-axes, fabric of this core is almost random and its average grain size is approximately 2 mm.

Fig. 1. Photographs of a thin specimen through crossed polaroids during compressive deformation test no. 2. Axial strain for each is: 1) 0%, 2) 3.4%, 3) 6.5%, 4) 10.0%, 5) 11.5%, 6) 12.4%. Compression axis is in the top and bottom direction of the figure.

Uniaxial compression tests of thin specimens

Processes of the c-axis rotation were recorded on photographs of thin specimens intermittently taken on the process of uniaxial compression tests by transmitted light through the crossed polaroids which sandwiched the specimen. Thin sections of approximately 2 mm revealed clear images of individual grains and even of slip bands when they appeared by stress, as can be seen in Figure 1. Every compression test was carried out using a testing machine (for details of which see Higashi and Shoji 1979) under an atmospheric pressure or 20 MPa hydrostatic pressure with a constant strain rate in the order of 10^-7s^-1. Experimental conditions including temperatures and the characteristics of the used specimens are given in Table 1. The thin specimen was also sandwiched by transparent acryllic-resin boards to avoid the buckling which may take place when such a specimen is compressed perpendicular to the thin section. This arrangement gave plane strain conditions.

Since it is expected that the rotation of the c-axis in an individual grain is related to internal strain, the internal local uniaxial strain of each grain was determined by the relative movement of several air bubbles in it (black dots in Figure 1). The least-square method of evaluating the plane strain using several data is developed from the standard method (Nye 1957a).

Uniaxial compression lest of bulk specimens

Uniaxial compression tests with bulk specimens for the study of fabric change were carried out with constant strain rate of 4.38 x 10^-7s^-1, under an atmospheric pressure. Experimental temperatures were -13 C (test 5) and -20 °C (test 6). When the total strain reached 10%, the test was terminated; the specimen was immediately removed from the testing machine and horizontal thin section specimens were prepared . Such procedures were carried out in a -20 °C cold room.

Rotation of c-axes

Figure 1 shows six successive photographs of a thin section specimen taken before the deformation and when the total strain of the specimen reached 3.4% 6.5%, 10.0%, 11.5% and 12.4% respectively in no. 2 test (Table 1). The slip bands appear in individual grain even when the total strain reached only 2% and they became clear as the strain increased. Since the c-axis must be perpendicular to the slip plane in each grain and it lay parallel to the plane of the wide area or to the plane of the figure, we can directly determine the c-axis of individual grains from these photographs. Angle x between the specimen’s compressive axis and the direction of the c-axis in each grain was measured using these determined c-axis direction at various increasing value of strain. We will call x the misorientation angle of a grain. Decreasing tendency of x with increase of the strain can be clearly seen in Figure 2 in which several examples of the change of x from different values x of various grains are plotted against the total strain E of specimens. They indicate that the c-axis rotates in individual grains as the deformation proceeds and the larger the value χ₀ the more rapid the rotation rate.

Table 1.

Fig. 2. Changes of the misorientation angle x with increasing strain ε. Several grains of different initial angle x₀ are designated by different symbols. Solid lines show the relationship x vs. ε by the equation 3.

Fig. 3. Change of the nearly random fabric of Dye-3 core no.120 a) before the compression test, b) after test no. 5, c) after test no.6, and d) the simulated fabric diagram by equation 4, starting from a) and strained 10%. The center of each diagram coincides with the compression axis.

Change of the random fabric

Fabric diagrams for the originally undeformed, and deformed specimens in the tests at -13 C and -20 C are illustrated in circles of three diagrams of Figure 3. This figure shows clearly that the c-axes become concentrated after the deformation, a tendency quantitatively expressed in the following way. Accumulated fraction curves as shown in Figure 4 are drawn by plotting the accumulated fraction F(e) of number of c-axes within certain misorientation angle θ against the values of θ divided into S intervals in this case. Three curves in Figure 4 exhibit F(e) for three data sets corresponding to Figure 3(a), (b) and (c). The almost straight solid line for the initial random fabric change to those broken one for test no 5 and dotted one for test no 6. They exhibit a steep slope below 40° and gentle slope at larger angles than 60°, indicating the tendency that the fabric becomes concentrated toward x approaches zero as the total strain increases. This process will be discussed and simulated in the next section with the aid of a formula obtained from the c-axes rotation process in the preceding subsection.

italic>Rotation of c-axes

In the case of the compression test of a single crystal under restraint, it is well known that the normal to a slip plane rotates as it tends to become parallel to the compression axis (Schmid and Boas 1935). Since the basal plane is a dominant slip plane in ice crystal, the angle x between the c-axis and the compression axis

Fig. 4. Cumulative fraction F(θ) curve for the fabrics of Dye-3 #120 before the test (----- ), after test S (- - - ) and test 6 ( . . . ).

decreases from its initial value compressive strain e equation:

(1)

Individual grains in our thin specimens are considered to be restrained by their neighbours when they are deformed. Therefore, it is expected that the rotation of the c-axis of each grain could be correlated to the local compressive strain e_ in the grain in such a way that ε is inserted for ε_S in eq. (1). ε_g in individual grains may be proportional to the gross axial strain ε of the polycrystalline plane specimen with a proportional constant as a function of X_o Figure 5 shows relationships between the measured ε_g of individual grains and their initial orientation angles Xo at two stages when the total axial strain reached 3% and 7%. Although data scatter considerably, they fit to the curve of an equation of the form:

Fig. 5. Relationship between the unixaial compressive strain ε_g of individual grains which had initial misorientotior. angle X₀ after a certain total strain, 3% and 7%. Solid lines show equation 2.

(2)

This equation implies that the arithmetic mean of ε_g over; X_o from O° to 90° is equal to the total axial strain £. Details of this implication together with its expansion to the three-dimensional case will be given elsewhere.

Inserting ε_g of this equation for ε_s in eq. (1), we obtain the changed orientation χ related to the total axial strain e and the initial angle ε of the orientation in the following form:

(3)

The solid lines in Figure 2 are the curves calculated by this equation (3) started from several certain initial χ₀ ^s. Good coincidence of the curves on the plotted empirical data of x vs. strain suggests that the rotation mechanism of c-axes envisaged to obey equation (I) is valid.

Formation of the broad single maximum fabric

As was described in the preceding subsection, the nearly random fabric of ice at comparatively shallow core of Dye-3 tends to concentrate into the broad single maximum when its bulk specimen was uniaxially compressed. In view of the fact that the c-axes of individual grains rotate so that the misorientation angle tends to decrease to zero degree in a thin polycrystalline specimen during the compressive deformation as described above, it is inferable that the same mechanism takes place for the concentration process of the c-axes in the bulk polycrystalline ice. Therefore, it can be expected that the broad single maximum fabric will be produced in nature when polycrystalline ice body is subjected to the compressive deformation as the case in the upstream accumulation zone of an ice sheet.

For predicting the degree of c-axes concentration owing to the deformation, two computer simulations on the formation process of the broad single maximum fabric were carried out. The constitutive equation used for these simulations is as follows:

(4)

This equation is of the same type as equation (3) because it should express the decreasing tendency of the misorientation angle θ with increasing strain. The angle θ is equivalent to x in equation (3) but express the angle between the c-axis of individual grain and the compressive axis in a bulk polycrystalline specimen. Since the fabric pattern varies with increasing strain, the simulation should be performed stepwise with a certain strain interval Δε. The constant Π in equation (3) is substituted by a parameter A which also varies with change of the fabric pattern.

The value of A can be determined by the following equation which is derived from an empirical fact that the uniaxial strain of a bulk specimen e is equal to the arithmetic mean of the uniaxial strain e of whole individual grains:

(5)

where N is the total number of grains of which misorientation angle e0, are measured. Angle Θ in equation (4) is the initial value of θ at each step of the simulation: the real initial value for each grain at the start and the resultant of the preceding step for the next step. Different values of A near 3 according to equation (5) were used in successive steps. In practice for the present, Δε was taken as 2%,

The one simulation started from the fabric of Dye-3 no. 120 core which exhibited nearly random fabric as illustrated in Figure 3a. Fabric diagram shown in Figure 3d is the resultant fabric when ε reached 10%. It is comparable to those obtained by compression tests as illustrated in Figures 3b and 3c. Another simulation was carried out starting from an artificially-made random fabric as shown in Figure 6a in which 200 orientation of c-axes were selected by the use of random number generated in the computer. Results of the simulation at 10%, 30% and 50% of the total compressive strain are given in Figures 6b, 6c and 6d.

Fig. 6. Simulated fabric diagrams started from artificially made random fabric with various degrees of uniaxial compressive strain. The compressive axis is in the center of each diagram.

Fig. 7. F(θ) - θ curve for the simulated fabrics shown in Fig. 6.

F(θ) curves, the accumulated fractions of the c-axes within θ are drawn in Figure 7 from such calculations as performed for the above simulation. The figure clearly shows that the broadness of the c-axes concentration decreases as the total strain increases. If we draw a horizontal line at a level of 50% of F(a) in Figure 7, its intersections with the predicted curve of F(θ) at various total strain ε give us the angle θ or the inclination from the compression axis at which the number of grains are halved in the fabric diagram. This is called the median inclination θ_m and a good indicator of the degree of concentration of the c-axes in the fabric. The strain dependence of θ_m obtained from results of the above simulation is given as a solid line in Figure 8.

Fig. 8. Relationship between the median inclination θ_m and the vertical strain for the Dye-3 core ice and specimens of compression tests. The solid line shows the strain dependence of θ_m obtained from results of simulated fabrics.

Individual relationships between the median inclination θ_m for the observed fabrics of specimens of ice core from various depths at Dye 3, Greenland and their given vertical strain ε are plotted on this figure (closed circles). The data for specimens of compression test no 5 and no 6 are also plotted (open circles). The vertical strain given to the ‘ice at a certain depth was estimated by a simple calculation based on Nye’s flow model (Nye 1957b), assuming that the ice thickness and the accumulation rate has not changed during the past few thousand years. The accumulation rate used for the calculation is the measured value of 0.54 ma^-1 at Dye 3 (Reeh and others 1978). For specimens of the uniaxial compression tests, the vertical strain was given by adding the compressive strain of 10% to the calculated vertical strain of 10% corresponding to the depth 201 m of specimen no. ¡20.

Field data and empirical results plotted on the figure fit fairly well with the solid line. This proves that the formation mechanism of the broad single maximum of the fabrics of ice cores from the depth of plateau area of Greenland ice sheet is principally the rotation of c-axes in individual grains composing the ice in it. This is believed to be generally true in polar ice sheets in which recrystallization does not occur intensively because of low temperatures.