~ec~on~~~~s~~s~ 48 (1978) Tl-T6 Tl o Elseviex Scientific Publishing Company, Amsterdam - Pxinted in The Netherlands

Letter Section ^__ ~-.-.-- --..-

Ultimate locking angles for conjugate and monoclinal kink bands

MICHAEL COLLIER

ABSTRACT

Collier, M., 1978. Ultimate locking angles for conjugate and rn~n~~lin~ kink bands. T~tan~physics, 48: TX--T6.

Honea, Reches and Johnson have int~duced a theory of conjugate and monoclinal kinking in rocks. A correction is required in expressions for the ultimate loeking angles of kink bands.

Honea, Reches and Johnson (Honea and Johnson, 1976; Reches and Johnson, 1976; and Johnson, 1977) have developed a theory of conjugate and monoclinal kink folding in multilayers characterized by strength of contacts between layers, and by flexible layers. They suggest that kinking is a result of buckling and of unstable yielding of contacts. Three conditions must be satisfied in order to produce kink bands:

(1) Principal stress difference, u 3-u1, is required to drive the buckling of the multilayer and provide shear stress at interfaces of the layers.

(2) A factor, K1, defined by Honea and johnson (1976), must be greater than zero.

(3) The angle, 0 I, between the direction of rn~~urn compression and the direction of a local pe~urbation of layering% must be less than a certain value in order for there to be unstable yielding of contacts:

[&‘I< 45”~$/2; @‘=B i-01 0)

where QI is the angle of burns friction of contacts between layers, and 6’ and cy are defined inFig, 1.

If the contacts between layers are strictly frictional (cohesion is zero) and if pore pressures are negligible, a particularly simple relation can be derived between the principal stresses and 8’:

@J/U1 =tan ~$'~#~/~~; (21

Here u3 is maximum compression.

‘I’2

Fig. 1. Kink band in a multilayer. o = angle of layering within kink band relative to non-folded layering, measured counterclockwise. 01 = angle of maximum compression direction relative to non-folded layering, measured clockwise. p = angle of kink edge relative to non-folded layering.

Stable yielding requires progressively higher u3/u1 values to maintain equilibrium during rotation of layers within a kink band. Unstable yielding, on the other hand, requires progressively lower UJU 1 values to maintain equilibrium. Suppose that a multilayer with a slight perturbation in slope of layering were subjected to a critical stress ratio, h, required for yielding at contacts (Fig. 2). The layering in the perturbation (with an initial 0 ’ value of point A, Fig. 2) would yield unstably, the stress ratio would de- crease, and the layers would rotate through higher values of 6’ into a stable configuration (point B, Fig. 2). In that configuration, the layers within the deformed band could again support the initial stress ratio, k. The value of 8’ indicated by point B, Fig. 2, becomes the “locked” orientation of layering within the kink band.

Fig. 2. Relations between ratio of principal stresses, a,/~,, and local layer inclination to direction of maximum compression. Eq. (2) is plotted here for @ = 30”. Arrows indicate stress paths as layers within the kink band rotate. (From Reches and Johnson, 1976.)

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The analyses discussed thus far are correct. There are errors, however, in the expression for the ultimate locking angle of layers within the kink band. The ultimate locking angle is defined as the angle, 19 ’ of eq. (2), for which the stress ratio approaches infinity. It is the maximum angle of inclination of the layering within the kink band relative to the direction of maximum com- pression. Where the maximum compression is parallel to non-folded layering, (CV = 0), the ultimate locking angle, according to eq. (2), is:

e L = +(90”-@) (3)

and not the value reported by Reches and Johnson (1976, p. 322) and John- son (1977, p. 270). Further, if the direction of maximum compression is inclined to non-folded layering, the ultimate locking angle is:

e;l = +(90”-G); 8’ = e + (Y (4)

and not the value reported by Reches and Johnson. Thus, the correction is twofold: aL should be replaced by ok, and G/2 by $ in their eq. (10).

These corrections affect the results presented by Reches and Johnson. The zones of unstable yielding are reduced to subtend angles of (45”-G/2), as shown in Fig. 3. Also, the ultimate locking angles reported in their fig. 13 should be corrected, as in Fig. 4 of this paper. For a material with a friction angle of 30”, this correction could account for a 15” shift in estimates of the principal stress directions.

The corrections also affect calculations of the ultimate slope angle, pL, of the edges of the locked kink band relative to non-folded layering. The

Fig. 3. Zones of yielding and no yielding for u3/u, --f a. Zone I - unstable yielding. Zone II - stable yielding. Zones III and IV - no yielding. Figure drawn for @ = 30”. (After Reches and Johnson, 1976.)

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L

+

Fig. 4. Relations between ratio of principal stresses, a,/~,, and slope angles of layers locally, 8, for various friction angles, @, for multilayers with purely frictional contact strengths. (After Reches and Johnson, 1976.)

angle p is defined in Fig. 1. Equation (7a) of Reches and Johnson (1976, p. 317) and Johnson (1977, p. 265) shows that:

p = ?(90”-8/2) (5)

From eq. (3), it follows that:

pL = ?(45” + @/2) (6)

when maximum compression parallels non-folded layering. In the general case, where u3 is inclined to non-folded layering by an angle, (IL, using eq. (4):

/3L = k(45” + @/2--a/2) (7)

and not the value reported by Reches and Johnson (1976, p. 331) and John- son (1977, p. 279).

A correction is also required in the expression for principal stress ratios for materials with both cohesive and frictional contact strength between layers:

jo;/o;l = tan(16j+#)/tan181+ (6)

[2Cl Io;I l/[W2lel) (I---tanI@ Itan@)

and not the value reported in eq. (13) of Reches and Johnson (1976, p. 328) and Johnson (1977, p. 276). With this correction, we see that the ultimate locking angles derived from eqs. (4) and (8) are identical. Thus, the ultimate

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locking angles of kink bands in a multilayer are independent of whatever cohesive strength the material may possess.

Finally, I would add a footnote to the analysis of kink bands by Reches and Johnson. Failure by faulting in an isotropic material, according to the Coulomb criterion, will occur along planes oriented +(45”-G/2) to u3 (as in Fig. 5A). But the over-all sense of failure by kinking in a multilayer subjected to layerparallel compression, from eq. (6), will be along planes oriented +(45” + G/2) to u3 (as in Fig. 5B). Inability to distinguish between Coulomb fractures and kink folds could lead to a 30” error in calculations of principal stress directions for a material with a friction angle of 30”.

Fig. 5A. Coulomb fracture, oriented (45’-@/2) to maximum compression.

Fig. 5B. Kink band, with edges oriented (45’ + G/2) to maximum compression.

ACKNOWLEDGEMENT

Much of this paper was developed during conversations with Arvid John- son. His insights have been (and continue to be) greatly appreciated.

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REFERENCES

Honea, E. and Johnson, A.M., 1976. A theory of concentric, kink and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers. IV. Development of sinusoidal and kink folds in multilayers confined by rigid boundaries. Tectonophysics, 30: 197-239.

Johnson, A.M., 1977. Styles of Folding. Elsevier, Amsterdam, 406 p. Reches, Z. and Johnson, A.M., 1976. A theory of concentric, kink and sinusoidal folding

and of monoclinal flexuring of compressible, elastic multilayers. VI. Asymmetric folding and monoclinal kinking. Tecto~ophysics, 35: 295-334.