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Discussion

ORIENTATION OF MINERAL LINEATION ALONG THE FLOW DIRECTION IN ROCKS: A DISCUSSION

N.C. GAY

Department of Geology, Imperial College, London (Great Britain)

(Received May 17, 1966)

The motion of individual particles during laminar shear is the mechanism proposed by Bhattacharyya (1966) to explain observed relationships between preferred orientations of minerals and flow directions in rocks. In the opin- ion of the present writer, however, this analysis does not explain one of the cited field examples, namely, lineations in deformed and metamorphosed rocks.

Describing these rocks, Bhattacharyya reports that a strong mineral lineation, lying in the axial plane schistosity, is paralleled by the major axes of deformed pebbles. The schistosity plane is interpreted as the slip plane of the shear zone and the direction of pebble elongation as that of major flowage in the rocks. If the rocks had been subjected to laminar shear, as Bhattacharyya suggests, these two directions could not possibly be parallel, except after an infinite amount of shearing strain. This is apparent from consideration of the geometry of simple shear (which is equivalent to the laminar flow of a viscous medium) as shown in Fig.1.

Fig.1. Deformation of a circle by simple shear. A. Initial position. B. Posi- tion after shearing. Unit of shear: y = tan q. Orientation of major axis of the strain ellipse:

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560 K.C. GAY

If it is assumed that during deformation in certain environments, rock can be considered as an extremely viscous fluid and that the component minerals or pebbles also behave as viscous bodies, then a pebble with the same viscosity as the surrounding medium will deform as a strain ellipse, as in Fig.1.

However, if the pebble is rigid, it will rotate about its centre during the shearing with an angular velocity given by Bhattacharyya’s eq.1 and will not achieve a stable orientation. At any particular time, the majority, bd certainly not all, of the pebble long axes will be oriented parallel to the shear direction.

If the pebble is not rigid but has a viscosity coefficient greater than that of the surrounding medium, it will deform and rotate simult~eously, the rate of rotation being between that of the strain ellipse and the rigid body. Taylor (1934) analysed this situation for very small fluid drops sub- jected to surface tension forces and deforming in a laminar shear field. Under these conditions, the deformed drops align themselves parallel to the shear direction. However, with particles of pebble dimensions, surface tension forces are no longer significant and the present writer has shown that deforming drops will rotate through the shear direction in the same manner as rigid bodies. Once past the shear direction, the forces acting on the particle will cause it to become less eccentric (i.e., the particle will tend to become circular). Note, also, that theoretically no deforming forces act parallel to the shear direction and therefore grains aligned parallel to this direction cannot become elongated as suggested by Bhattacharyya.

Therefore, if the mineral lineation in the rocks described by Bhattacharyya is parallel to the shear direction and laminar flow has been the deforming mechanism, some of the pebbles in the rocks will lie at an angle to the lineation direction. From the evidence given in the paper, this is apparently not the case and hence it can be concluded that the mechanism proposed by Bhattacharyya is incorrect.

It seems to the present writer that a pure shear deformation is better able to explain the mineral and pebble orientations.

The equations:

define a pure shear plane strain in which the intermediate axis of the strain ellipsoid remains constant. The symbols zt, c and zt! represent the velocity components parallel to the X-, Y- and Z- axes in Fig.2 and i = de/dt is the rate of natural strain, E , as defined by Jaeger (1962, p.68).

Jeffery’s (1922) method for solving the motion of rigid ellipsoidal particles can be applied to this type of flow and the writer has derived the following equations to define the changes in position of an ellipsoid of rev- olution with axes CI : b : b, during pure shear:

--&J$ =_a2-b2 ; a2 + 02

-26 sin 20

= $1 cot 2qo’

which are equivalent to eq.1 and 2 of Bhattacharyya (1966). The angles @ and p’, the complement of Bhattacharyya’s angle cp, are defined in Fig.2. Elim- inating the time factor from these equations, they can be written:

Tectonophysics, 3 (6) (1966) 55%j64

MINERAL LINEATION ALONG THE FLOW DIRECTION IN ROCKS 561

Fig.2. Orientation of the major axis of the ellipsoidal particle in the pure shear field u = &L, v = -<y, w = 0.

-2dB sin 28 =

cot 2cp’dp’

(2d

(W

Integrating eq.2a gives:

In tan cp’ -lntancpti +-$-$ (-2E) f-

where the subscripts i and f indicate the initial and final angles respectively. In terms of ,q (= 90” - q’) and the axial ratio of the strain ellipse for the pure shear deformation:

ln$$ =2~ r the above equation becomes:

a2 - b2 In cot qf = In cot ‘pi + - In A2

a2 + b2 r 7 (4)

and relates the change in orientation 9, the shape of the particle and the amount of applied strain.

Similarly, the variation in 8 with p is obtained by integrating eq.3a toget:

tan ef sin Qi

tG7i = r- sin +f (5)

If a particle lies parallel to either of the OX- or OY- axes of deforma- tion (i.e., C+J = O” or 900), there is no change in orientation but 0 varies according to the equation:

lntanBf=lntanQi+~ln~ r The significance of the equations is as follows: (1) Any ellipsoid lying in the XY deformation plane with its axes

(6)

Tectonophysics, 3 (6) (1966) 55% 564

N.C. GAY

parallel to the strain axes, will remain stationary in these stable positions. (2) If it lies at an angle cp to the strain axes it will rotate towards the

direction of elongation OX becoming parallel to it after an infinite amount of deformation.

(3) If the particle does not lie in the deformation plane but has its major axis parallel to either of the strain axes, then it will not rotate but the angle 9 will either decrease (if cp = O”) or increase (if cp = 90“) according to eq.6.

(4) If the particle lies in a general position with its major axis at angles 50 and B to the strain axes and the deformation plane, then during pure shear it will rotate towards the direction of elongation (i.e., cp increases) while 0 decreases until cp = 45”, beyond which it increases.

Thus the major axis of the ellipsoid tends to become parallel to the major strain axis and to lie in thedeformation plane.

In other words, with increasing deformation the degree of preferred orientation along the direction of elongation in the deforming material in- creases. The fact that the mathematical analysis precludes the particle axes from reaching the stable positions is unimportant, since a high degree of sub-parallelism between major axes and the flow direction can be achieved withilz relatively small amounts of strain. Moreover, the effect of particle interactions will be to enhance the preferred orientation.

If the particles are not rigid but change shape and rotate simultane- ously during the deformation, the mathematical analysis becomes far more complicated. However, the present writer has solved the two-dimensional problem numerically and the results show that a significant degree of par- allelism of particle axes is rapidly obtained.

The results described in this discussion have been determined by the writer while working on the geological applications of the deformation of inhomogeneous materials, details of which are to be published later.

ACKNOWLEDGEMENTS

Dr. N. Price and Dr. D. Elliott critically reviewed the manuscript. The writer gratefully acknowledges receipt of a grant from the South African Council for Scientific and Industrial Research.

REFERENCES

Bhattacharyya, D.S., 1966. Orientation of mineral lineation along the flow direction in rocks. Tectonophysics, 3(l): 29-33.

Jaeger, J.C., 1962. Elasticity, fracture and flow, 2nd. ed. Metheun, London, 208 pp. Jeffery, G.B., 1922. The motion of ellipsoidal particles immersed in a viscous fluid.

Proc. Roy. Sot. (London), Ser. A, 102: 161-179. Taylor, G.I., 1934. The formation of emulsions in definable fields of flow. Proc. Roy.

Sot. (London), Ser. A, 146: 501-523.

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