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An important aid to the geologist in analyzing rock deformation is the rock particles which contain the stress history of the rock, in the form of strain. Analysis of such particles may enable the geologist to determine the finite strain, initial shape and orientation of the particles. The ideal strain marker is an originally spherical body which has the same ductility as its host, because itâ€™s final shape is that of the strain ellipsoid itself and this will determine the state of strain. Robin (1977) defines a strain marker as follows, “Strain markers are geologic bodies within a rock which, during the deformation of that rock, have retained their identity but did not differ from their surrounding material in their mechanical behavior.” Cloos (1947) study of deformed ooids came very close to the ideal state, however, there are departures from the ideal state and these may be of two kind: (1) there may be ductility contrast between the strain marker and the host rock, (2) there is no ductility contrast but the initial state of the strain markers is not spherical. In the case of conglomerates, the shape of the constituent particles is generally non-spherical, sub ellipsoid, but may vary. The ductility contrast will vary between the situations where there is no ductility contrast to where there is an extreme ductility contrast. Therefore, there can be many problems encountered in the strain analysis of conglomerates.

### No Ductility Contrast

Gay (1968) has shown that the viscosity ratio between a particle and the mean viscosity of a particle/matrix system is dependent on the relative population of particles and matrix present. Thus, in conglomerates, where the proportion of pebbles to matrix is high, the ductility contrasts are probably low; therefore, assuming the ductility contrasts are negligible it should be possible to determine the following: (a) the initial shape of the particle; (b) the initial axial orientation; (c) the strain intensity; and (d) the strain orientation.

Since the initial shapes of particles within the conglomerates are unknown simple finite-strain analysis is not possible. Analysis of such particles requires the following assumptions: (1) an adequate number of particles in the rock having a variety of shapes; (2) an initially isotropic fabric so that the particles were randomly orientated; and (3) the rock underwent no volume change. In the treatments Ramsey and Dunnet (1969) use, a further assumption is necessary. This is that the initial shape be ellipsoid. Robin (1977) has illustrated a method where it is not necessary fore the particle shape to be ellipsoid.

### Dunnet’s Method

Dunnet’s (1969) method of strain analysis involves plotting an Rf (final deformed particle axial ratio) vs f (angle from the Rf long axis to the maximum principle strain direction or reference direction) graph. The procedure adopted to determine the finite strain value (Rs) is as follows.

Firstly, the mean of both the Rf and f axis is determined. The angular mean should coincide with the orientation of the principle strain trace and with the maximum and minimum plotted Rf values. The mean of the Rs values does not coincide with Rs but will be a slightly higher value. The intersection of mean v and logarithmic mean Rf is considered as the finite strain value (Rs). The number of points in each of the four quadrants is counted and should be equal. If not, they define one type of asymmetry. Lack of coincidence of f mean with the maximum and minimum Rf values define a second type and lack of coincidence of f mean and the principle trace is a third type of asymmetry. If the diagram is reasonably symmetrical, the appropriate Rf/v curves are filled to the data. A number of such curves have been determined by Dunnet for a given Rs value. The curves are adjusted to the visual best fit. For asymmetric diagrams, or where initial orientation of particles in bedding is suspected, the position of the bedding trace must be considered. Particles with their initial long axes at a high angle to the maximum strain axis will produce Rf/f ratios less than the strain ratio (Ri). Particles at a low angle to the maximum strain direction will produce Rf/f ratios greater than the strain ratio. Strain ratios from the three principle planes xy, yz and xz can be compared and, if not consistent, may be recalculated.

The main factors which affect the interpreted value of Rs are the effects of: (1) initial orientation of particles; (2) superimposed deformation; (3) rotational strain; (4) ductility contrasts; and (5) inhomogeneous strain over the specimens analyzed. Initial preferred orientation and superimposed deformation produce artificially similar results. Ramsay (1967) has illustrated that an initial planar oriented fabric produces an Ri/f distribution to Rf/f curves. Thus, modification of an initially preferred oriented fabric is similar to the modification of a fabric produced by earlier deformation. In the case of an initial oriented fabric which has undergone deformation the final fabric is generally asymmetric. However, if the initial fabric is perfectly planar the Rs value, as in the case of a superimposed fabric, is the finite product of the initial and secondary fabric.

In an example, Dunnet cites that it is impossible to determine whether two strains have produced a fabric unless additional evidence, such as cleavage oblique to the total finite strain orientation is present. He also cites that perfect initially planar fabrics are extremely uncommon; therefore, the finite strain orientation of an initially oriented fabric will almost always be asymmetric.

Rotational strain similarly produces a finite strain oblique to the axes and it will also be less than the product of the strain increments. Therefore, finite strain can be derived for both irrotational and rotational strain.

The presence of inhomogeneous strain on the scale of a single thin section is uncommon. If present, the sample should be avoided. Inhomogeneous strain can be recognized by visual inspection since a gradational variation in the strain intensity necessitates a variation in the strain orientation. Therefore, divergence of cleavage may indicate inhomogeneous strain.

In summary, comparison of the deformed fabrics of initially non-spherical sub-ellipsoid particles with a theoretical analysis of ellipses deformed by homogenous strain allows determination of the finite strain and initial axial ratio. Samples of several particle lithologies should be avoided or measurement restricted to particles of a single lithology. Similarly, specimens showing initial preferred orientation in bedding should be avoided or treated with caution. Furthermore, samples collected from areas of rotational strain or superimposed deformation should be analyzed with care.

### Robin’s Method

Ramsay (1967, Dunnet (1969) and Shimamoto and Ikeda (1976) have all developed methods to obtain strain when the strain markers are assumed randomly oriented and are ellipsoids of unknown initial axial ratio. Robin (1977) has removed one of these restrictions in order to calculate strain; that is, the strain markers are not necessarily ellipsoid but may be of any shape. Of course, ellipsoid particles may also be used. Robin’s method requires the following assumptions: (1) an adequate number of particles within the rock having varied shape; (2) no ductility contrast between the particles and host rock; (3) an initially isotropic fabric so that the particles are randomly oriented; and (4) the rock underwent no volume change. These assumptions are less restrictive than those of Ramsay and Dunnet.

Robin’s treatment is restricted to two dimensions, however, results obtained from several planar directions may be combined into a single three dimensional model. Robin designates the maximum and minimum strain directions as l 1 and l 3 respectively. Through the centre of gravity, c, of each unstrained marker, two diameters, parallel to the 1 and 3 directions, can be drawn. Therefore, in each marker, j, a diameter aj, parallel to the 1 direction, and a diameter cj, parallel to the 3 direction are present. In the strained state, these diameters are defined as aj’ and cj’. Robin mathematically proves the following:

(sum from j=1 to n) log (aj’/cj’) n log (sqr(l 1/l 3))

Therefore, the diameter aj’ and cj’ are measured on n strain markers, the ratios aj’/cj’ calculated, and the axial strain ratio is given by their logarithmic average. The method requires that the strain directions of 1 and 3 be known. They could be found by evaluating the equation for many directions and choosing the direction for which its value is the maximum. However, it is simpler and probably sufficient to estimate the direction of maximum principle strain from the trace on a planar section of a foliation, or from a histogram of directions of the longest diameter of each deformed marker.

Robin says, ” A difficulty arises from the fact that the center, c, for each marker must be determined in the strained state rather than in the undeformed one. He goes on to say, ” However, the center of gravity is not easy to determine, except for centrosymmetric markers, such as ellipsoids. Other rules to choose a center, such as the midpoint between two points that are the farthest apart on the contour of the marker are much easier to use. However, because this determination must be made in the strained state rather than in the unstrained state, and because two markers which were initially identical except for a rotation, are no longer identical in the strained state, the centers determined by such rule would not coincide in the two “destrained” markers”, and, ” For most markers it is unlikely that choosing centers with such a simple rule would lead to significant errors in practice, as a/c is generally not very sensitive to the exact location of c.”

If the assumptions made earlier are upheld then only one major problem in the strain analysis of conglomerates becomes apparent. That is, like most methods of strain analysis, only the total strain will be determined in situations where an initial preferred orientation of particles in bedding or deformation is present. In the case of ellipsoid strain markers, a randomness test (See Robin, 1977) may be used to detect an initial fabric. For non-ellipsoid particles, it is impossible to determine an initial fabric; therefore, restricting the use of the method.

### Ductility Contrast

Gay (1968a) has shown that the viscosity ratio between a particle and the mean viscosity of a particle/matrix system is dependent on the concentration of the particles. Therefore, where the proportion of pebbles to matrix is low there will be a high ductility contrast. He has illustrated that an increase in the viscosity of an initial ellipsoid particle will greatly reduce the change in shape of the particle and the rate at which the long axis will move towards the maximum principle strain direction. The final Rf/f ratio will produce curves similar to those presented by Dunnet, but with the maximum fluctuation reduced and an Rs value which is much less than the deformation the matrix has undergone. A high particle to matrix ratio will have the effect of reducing the ductility contrast.

Gay (1968b) illustrates a method to determine finite strain in deformed conglomerates. This method requires an estimation of the “viscosity ratio” between pebbles and matrix , and the initial shape of the pebbles. When Dunnet’s technique is used to calculate Rs values in particles of different lithology, the viscosity ratio can be determined without estimation of the initial particle shape by substituting the relative strain ratios, determined by Dunnet’s method, into equations of Gay.

The major problem associated with Gay’s method is that the initial pebble shape and orientation must be known to calculate viscosity ratios. In the calculations, Gay assumes the original shape to be spherical and orientation to be parallel to the axes of the strain ellipse. This results in a very approximate viscosity ratio and it is necessary to make as many pebble measurements as possible to refine the calculation.

### Conclusion

The problems associated with strain analysis in conglomerates are not unique. They are also encountered in the study of other rocktypes, such as oolites and pisolites. In general, the major problems affecting the determined value of finite strain are: (1) initial preferred orientation of fabrics; (2) superimposed deformation; (3) rotational strain; (4) ductility contrasts; and (5) inhomogeneous strain. Numerous techniques , discussed above, have been developed to overcome these problems. The end product of all these techniques is to determine the strain intensity and orientation. This permits us to interpret the fabrics of deformed rocks, the significance of cleavage and lineation, and to correlate strain magnitudes and fabric patterns.

### References

Cloos. E., 1947, “Oolite Deformation In South Mountain Fold, Maryland”. Geol. Soc Amer., Bull. 58, pp.843-918.

Dunnet, D., 1969, ” A Technique of Finite Strain Analysis Using Elliptical Particles”. Tectonophysics, V.7, pp. 117-136.

Dunnet, D. and Siddans, A.W.B., 1971, ” Non Random Sedimentary Fabrics and Their Modification By Strain”, Tectonophysics, V12, pp.307-325.

Elliot, D, 1970, “Determination of Finite Strain and Initial Shape from Deformed Elliptical Objects”, Geol. Soc. Amer., Bull. 81, pp.2221-2236.

Gay, N.C., 1968, “Pure Shear and Simple Shear Deformation of Inhomogeneous Viscous Fluids”, Tectonophysics, V.5, pp.211-234, pp.295-302.

Ramsay, J.G., 1967, “Folding and Fracturing of Rocks”, MacGraw-Hill, NY, 568p.

Robin, P.F., 1977, “Determination of Geologic Strain Using Randomly Oriented Strain Markers of Any Shape”, Tectonophysics, V.42, ppT7-T16.

Shimamoto and Ikeda, 1976, “A Simple Algebraic Method for Strain Estimation From Deformed Ellipsoid Objects”, Tectonophysics, V.36, pp.315-337.