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Natural indicators of solid-body rotation in deformed rocks
Tectonophysics, 63 (1979) T15-T20 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
T15
Letter Section NaturaI indicators of solid-body rotation in deformed rocks W.M. SCHW~RDTNE~
~epurt~e~t
of Geology, Universes of Toronto, Toronto, Ont. M5S 1Al ~Ca~ada~
(Submitted May 21, 1973; accepted for publication November 11,X978)
ABSTRACT Schwerdtner, W.M., 1979. Natural indicators of solid-body rotation in deformed Tectonophysics, 63 : T15-T20.
rocks.
Mechanically passive features with structural polarity such as faint graded beds or cross beds are apt to be the beet natural gauges of two-d~ension~ p~eo-rotation. If the primary dip direction of cross bedding is known, then the planes of bedding and cross bedding collectively record the magnitude of three-d~ension~ rotation. Rigid spherical inclusions fail to roll as a result of oblique superposition of pure finite strains or due to bulk rotation of iithologic units. In progressive simple shear, the rigid-body rotation of spherical or circular inclusions is generally greater than the solid-body rotation of the weak matrix, which cannot exceed 90 degrees.
INTRODUCTION
As shown in several recent papers (Ramsay, 1969, 1976; Schwerdtner, 1977; and Cobbold, 1977), restoration of the originaI shape of a beterogeneously deformed body requires that the ro~~~~~~~strain be known at a sufficient number of points. Unfo~unateIy, natural gauges of solid-body rotation are rarely found in deformed rock bodies. In his comprehensive paper on rotated garnets, Rosenfeld (1970, sections 1 and 4) advocates the exclusive use of spherical rigid inclusions as indicators of solid body rotation. He rejects the classical concept of defining solid-body rotation as the net change in orientation of the principal lines of homogeneous finite strain. For infinitesimal simple shear, the solid-body rotation of a Newtonian matrix is equal to the absolute rotation of spherical rigid inclusions (Jeffrey, 1923; Rosenfeld, 1970; Ghosh and Ramberg, 1976). This is not true for all other types of ~~~~~ strain, or for most states of finite strain. SOLID-BODY
ROTATION VERSUS SHEAR~G
STRAIN
Any general homogeneous strain involves a solid-body rotation and a pure strain (also called “irrotational strain”). Within a domain of general homo-
T16
geneous strain, most line elements rotate due to pure strain as well as solid-body rotation. (The strain-induced rotation of a line element, called angle of shear, is generally measured with respect to an initially perpendicular plane.) Only the principal axes of defo~atio~ escape a rotation due to pure strain; directed principal lines are therefore most suitable for recording the magnitude of solid-body rotation. Nevertheless, any non-principal line with known original orientation will be useful if its strain-induced rotation can be determined and subtracted from the net rotation. The original normal to graded bedding, the directed dip line on primary cross beds, and the trace of intersection between cross bedding and bedding are examples of such arbitrary lines (Schwerdtner, 19’76, fig. 2). In combination, these structural elements record the magnitude of three-dimensional solid-body rotation of sedimentary units. Solid-body rotation may result from the operation of three mechanisms: (1) progressive simple shearing*, (2) oblique superposition of pure finite strains, and (3) bulk rotation of entire domains. The first and second mechanisms involve a shift of the principal directions from three perpendicular line elements to other orthogonal line elements, in the course of deformation The third mechanism is characterized by the fact that the principal directions are pegged to specific orthogonal line elements, throughout progressive deformation (cf. Ramsay, 1967, fig. 7-63). The solid-body rotation associated with a general finite strain can involve all three mechanisms. It will be seen that the net rotation of small spherical inclusions is greater than that component of solid-body rotation which is due to components of progressive simple shearing. Nevertheless, syntectonic rigid inclusions can provide valuable info~ation about the actual kinematic path which has been followed in a~cumulat~g a given rotational strain. FINITE AND INFINITESIMAL
ROTATIONS
IN CONTINUUM
MECHANICS
The displacement gradient matrix which represents a general state of finite homogeneous strain can be decomposed into, or regarded as the product of, two displacement gradient matrices, one specifying a solid-body rotation and the other a state of pure strain (Biot, 1965; Elliot, 1970; Cobbold, 1977). While the pure-strain matrix changes with changing sense of decomposition, the rotation matrix is unaffected by that sense (Fig. 1). It is customary in mechanics to use the right-polar decomposition (first accumulating the pure strain and then pe~o~~g the solid-body rotation). This decomposition is also useful in restoring the original geometry and orientation of st~t~phi~ units and graded beds in symmetrical folds (Schwerdtner, 1976, fig. 2 and 1977). The finite-rotation matrix cannot be obtained by splitting the displacement gradient matrix into its symmetrical and antisymmetrical (also called skew-symmetrical) parts (Ramsay, 1967, p. 124). As shown by Love (1944, *Coaxial superposition
of a series of infinitesimal
simple shears,
.;... .: :...:: -rl,
HJ
T17 Y
k
y
--cxv---q
.:.:::. ...
:::..:,::: :CY .. :: : :: ::;;,.., .::,;;:. . .._..
I:::.‘,. ::..::
::
:
x
;
x
cyx
%4
*
original solid
square
-body
rotation
original square
solid
- body
rotation
pure
strain
Fig. 1. Different sense of decomposition of a biaxial finite deformation with transformation coefficients (cc); right-polar decomposition (A) and left-polar decomposition (B). Note that corresponding coefficients of pure strain are unequal, i.e. ati + bij. Points I and II are at (OJ) and (1,O) after rotation.
Y
X
Fig. 2. Geometric significance of an antisymmetric displacement-gradient dimensions (rotation and dilation of a unit square). Note elongation gonal (heavy line).
matrix w in two of the original dia-
T18
p. 70) and Nadai (1950, p. 110) the finite-rotation matrix can itself be decomposed into a cubic dilatation and a set of axial simple shears with the same sense of differential displacement (Fig. 2), represented by an antisymmetricaf matrix*. Only ~f~itesimal rotations, for which the dilatation is negligible, may be represented by antisymmetrical matrices alone. RIGID SPHERICAL SHEAR
INCLUSIONS
AS INDICATORS
OF PROGRESSIVE
SIMPLE
In the first section of his comprehensive paper, Rosenfeld presents an example of a homogeneous deformation that is attained via different kinematic paths (1970, fig. 1). Irrespective of kinematic path, this deformation involves the same solid-body rotation as recorded by the net shift of the principal lines of strain. But the contribution of progressive simple shear to the total rotation varies according to the chosen path, a fact which leads Rosenfeld (1970) to postulate that the solid-body rotation of a given deformation cannot be determined from the initial and final geometries. In theory, a state of finite simple shear can be produced without simple shearing. “Progressive simple shear” thus refers to a finite strain which has actually been attained by simple shearing. This distinction is important as only progressive simple shear leads to detectable absolute rotations of spherical rigid inclusions (Jeffery, 1923; Rosenfeld, 1970; Dixon, 1976; Ghosh and Ramberg, 1976). Only for some states of general finite strain can these absolute rotations be determined from microstructural data (Ghosh and Ramberg, 1978). Let us consider an inelastic matrix subjected to pure shear or homogeneous simple shear. The presence of rigid inclusions in the ductile matrix gives rise to strain perturbations, i.e. local fields of heterogeneous strain, which generally lead to solid-body rotations (Ramsay 1969, p. 52). Only if a perturbation has a symmetrical strain pattern (Fig. 3), will the inclusions retain their initial orientation. This is not the case for progressive simple shear, which invariably rotates all rigid spherical inclusions (Fig. 3B). Because the absolute rotation is chiefly dependent on the strain pe~urbation rather than on the overall simple shear of magnitude y, we find that, for a linear viscous fluid, the solid-body rotation of the unperturbed matrix (4,) differs from the rigid-body rotation of the small spheres ($,). According to Nadai (1950), Jeffery (1923), Rosenfeld (1970) and Ghosh and Ramberg (1976, in press), 4% = tan -’ (--r/2) whereas cpS= -y/2. Note that &, < b/21, but I&I can approach infinity** (no limit as to the number of revolutions). Nevertheless, *According to widespread usage (Nye, 1957; Ramsay, 196’7, p, 124), the antisymmetrical matrix is defined in such a way that all leading-diagonal terms are zero. This convention has not been followed in recent papers (for example, Cobbold, 1977, p. T2). **Dixon (1976, p. 102) put #m = q+, and apparently did not recognize that the two rotations can be vastly different.
T19
Fig. 3. Deformation 1975). Distortion of simple shear (B). The strain, at the onset of
pattern around rigid circular inclusions (modified from Ghosh, initially orthogonal grids due to pure shear (A) and homogeneous original grid lines were parallel to the principal directions of overall progressive deformation.
can be calculated from &, which must be determined from microstructural evidence (Rosenfeld, 1970; Dixon, 1976; Ghosh and Ramberg, in press),
Om
CONCLUSIONS
In summary, the following conclusions may be drawn: (1) The shift of the principal lines of finite strain, from their initial orientation to the final orientation, provides the best measure of finite solid-body rotation, either at a point or for a large domain of homogeneous strain. For small deformations, this shift corresponds to the antisymmetric part of the infinitesimal-displacement gradient matrix. (2) Mechanically passive, primary features with structural polarity are probably the best natural gauges of paleo-rotation. (3) Spherical “rigid” inclusions are rolled by simple shearing only, which is but one mechanism of solid-body rotation. (4) The ductility contrast between “rigid” inclusions and rock matrix gives rise to strain perturbations which can lead to solid-body rotations that differ from those of the adjacent unperturbed regions. ACKNOWLEDGEMENT
I would like to thank my colleague Pierre-Yves Robin for a critical review of the original draft.
REFERENCES Biot, M.A., 1965. Mechanics of Incremental Deformations. Wiley, New York, N.Y., 604 pp. Cobbold, P.R., 1977. Compatibility equations and the integration of finite strains in two dimensions. Tectonophysics, 39 : Tl-T6.
T20 Dixon, J.M. 1976. Apparent double rotation of phorphyroblasts during a single progressive deformation. Tectonophysics, 34: 101-115, Efliott, D., 1970. Determination of finite strain and initial shape from deformed elliptical objects. Geol. Sot. Am. Bull., 81: 2221-2236. Ghosh, S.K., 1975. Distortion of planar structures around rigid spherical bodies. Tectonophysics, 28: 185-208. Ghosh, S.K. and Ramberg, H., 1976. Reorientation of inclusions by combination of pure shear and simple shear. Tectonophysics, 34: l-70. Ghosh, SK. and Ramberg, H., 1978. Reversal of spiral direction of inclusion trails in paratectonic porphyrobl~ts. Te~tonophysics, 51: 83-97. Jeffery, G.B., 1923. The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Sot. Lond., 102A: 161-179. Love, A.E.H., 1944. The Mathematics Theory of Elasticity. Dover Publ., New York, N.Y., 4th ed., 643 pp. Nadai, A., 1950. Theory of Flow and Fracture of Solids. McGraw-Hill, New York, N.Y., Vol. I, 2nd ed., 572 p. Nye, J.F., 1957. Physical Properties of Crystals. Oxford University Press, London, 322 Rankly, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, New York, N.Y., 568 pp. Ramsay, J.G., 1969. The measurement of strain and displacement in erogenic belts. In: Time and Place in Qrogeny. Geol. SOC. London, pp. 43-79. Ramsay, J.G., 1976. Displacement and strain. Philos. Trans. R. Sot. Lond., Ser. A, 283: 3-25. Rosenfeld, J.L., 1970. Rotated garnets in metamorphic rocks. Geol. Sot. Am., Spec. Pap., 129: 106 pp. Schwerdtner, W.M., 1976. A principal difficulty of proving crustal shortening in Precambrian shields. Tectonophysics, 30: TlS-23. Schwerdtner, W.M., 1977. Geometric interpretation of regional strain analyses. Tectonophysics, 39: 515-531.
T15
Letter Section NaturaI indicators of solid-body rotation in deformed rocks W.M. SCHW~RDTNE~
~epurt~e~t
of Geology, Universes of Toronto, Toronto, Ont. M5S 1Al ~Ca~ada~
(Submitted May 21, 1973; accepted for publication November 11,X978)
ABSTRACT Schwerdtner, W.M., 1979. Natural indicators of solid-body rotation in deformed Tectonophysics, 63 : T15-T20.
rocks.
Mechanically passive features with structural polarity such as faint graded beds or cross beds are apt to be the beet natural gauges of two-d~ension~ p~eo-rotation. If the primary dip direction of cross bedding is known, then the planes of bedding and cross bedding collectively record the magnitude of three-d~ension~ rotation. Rigid spherical inclusions fail to roll as a result of oblique superposition of pure finite strains or due to bulk rotation of iithologic units. In progressive simple shear, the rigid-body rotation of spherical or circular inclusions is generally greater than the solid-body rotation of the weak matrix, which cannot exceed 90 degrees.
INTRODUCTION
As shown in several recent papers (Ramsay, 1969, 1976; Schwerdtner, 1977; and Cobbold, 1977), restoration of the originaI shape of a beterogeneously deformed body requires that the ro~~~~~~~strain be known at a sufficient number of points. Unfo~unateIy, natural gauges of solid-body rotation are rarely found in deformed rock bodies. In his comprehensive paper on rotated garnets, Rosenfeld (1970, sections 1 and 4) advocates the exclusive use of spherical rigid inclusions as indicators of solid body rotation. He rejects the classical concept of defining solid-body rotation as the net change in orientation of the principal lines of homogeneous finite strain. For infinitesimal simple shear, the solid-body rotation of a Newtonian matrix is equal to the absolute rotation of spherical rigid inclusions (Jeffrey, 1923; Rosenfeld, 1970; Ghosh and Ramberg, 1976). This is not true for all other types of ~~~~~ strain, or for most states of finite strain. SOLID-BODY
ROTATION VERSUS SHEAR~G
STRAIN
Any general homogeneous strain involves a solid-body rotation and a pure strain (also called “irrotational strain”). Within a domain of general homo-
T16
geneous strain, most line elements rotate due to pure strain as well as solid-body rotation. (The strain-induced rotation of a line element, called angle of shear, is generally measured with respect to an initially perpendicular plane.) Only the principal axes of defo~atio~ escape a rotation due to pure strain; directed principal lines are therefore most suitable for recording the magnitude of solid-body rotation. Nevertheless, any non-principal line with known original orientation will be useful if its strain-induced rotation can be determined and subtracted from the net rotation. The original normal to graded bedding, the directed dip line on primary cross beds, and the trace of intersection between cross bedding and bedding are examples of such arbitrary lines (Schwerdtner, 19’76, fig. 2). In combination, these structural elements record the magnitude of three-dimensional solid-body rotation of sedimentary units. Solid-body rotation may result from the operation of three mechanisms: (1) progressive simple shearing*, (2) oblique superposition of pure finite strains, and (3) bulk rotation of entire domains. The first and second mechanisms involve a shift of the principal directions from three perpendicular line elements to other orthogonal line elements, in the course of deformation The third mechanism is characterized by the fact that the principal directions are pegged to specific orthogonal line elements, throughout progressive deformation (cf. Ramsay, 1967, fig. 7-63). The solid-body rotation associated with a general finite strain can involve all three mechanisms. It will be seen that the net rotation of small spherical inclusions is greater than that component of solid-body rotation which is due to components of progressive simple shearing. Nevertheless, syntectonic rigid inclusions can provide valuable info~ation about the actual kinematic path which has been followed in a~cumulat~g a given rotational strain. FINITE AND INFINITESIMAL
ROTATIONS
IN CONTINUUM
MECHANICS
The displacement gradient matrix which represents a general state of finite homogeneous strain can be decomposed into, or regarded as the product of, two displacement gradient matrices, one specifying a solid-body rotation and the other a state of pure strain (Biot, 1965; Elliot, 1970; Cobbold, 1977). While the pure-strain matrix changes with changing sense of decomposition, the rotation matrix is unaffected by that sense (Fig. 1). It is customary in mechanics to use the right-polar decomposition (first accumulating the pure strain and then pe~o~~g the solid-body rotation). This decomposition is also useful in restoring the original geometry and orientation of st~t~phi~ units and graded beds in symmetrical folds (Schwerdtner, 1976, fig. 2 and 1977). The finite-rotation matrix cannot be obtained by splitting the displacement gradient matrix into its symmetrical and antisymmetrical (also called skew-symmetrical) parts (Ramsay, 1967, p. 124). As shown by Love (1944, *Coaxial superposition
of a series of infinitesimal
simple shears,
.;... .: :...:: -rl,
HJ
T17 Y
k
y
--cxv---q
.:.:::. ...
:::..:,::: :CY .. :: : :: ::;;,.., .::,;;:. . .._..
I:::.‘,. ::..::
::
:
x
;
x
cyx
%4
*
original solid
square
-body
rotation
original square
solid
- body
rotation
pure
strain
Fig. 1. Different sense of decomposition of a biaxial finite deformation with transformation coefficients (cc); right-polar decomposition (A) and left-polar decomposition (B). Note that corresponding coefficients of pure strain are unequal, i.e. ati + bij. Points I and II are at (OJ) and (1,O) after rotation.
Y
X
Fig. 2. Geometric significance of an antisymmetric displacement-gradient dimensions (rotation and dilation of a unit square). Note elongation gonal (heavy line).
matrix w in two of the original dia-
T18
p. 70) and Nadai (1950, p. 110) the finite-rotation matrix can itself be decomposed into a cubic dilatation and a set of axial simple shears with the same sense of differential displacement (Fig. 2), represented by an antisymmetricaf matrix*. Only ~f~itesimal rotations, for which the dilatation is negligible, may be represented by antisymmetrical matrices alone. RIGID SPHERICAL SHEAR
INCLUSIONS
AS INDICATORS
OF PROGRESSIVE
SIMPLE
In the first section of his comprehensive paper, Rosenfeld presents an example of a homogeneous deformation that is attained via different kinematic paths (1970, fig. 1). Irrespective of kinematic path, this deformation involves the same solid-body rotation as recorded by the net shift of the principal lines of strain. But the contribution of progressive simple shear to the total rotation varies according to the chosen path, a fact which leads Rosenfeld (1970) to postulate that the solid-body rotation of a given deformation cannot be determined from the initial and final geometries. In theory, a state of finite simple shear can be produced without simple shearing. “Progressive simple shear” thus refers to a finite strain which has actually been attained by simple shearing. This distinction is important as only progressive simple shear leads to detectable absolute rotations of spherical rigid inclusions (Jeffery, 1923; Rosenfeld, 1970; Dixon, 1976; Ghosh and Ramberg, 1976). Only for some states of general finite strain can these absolute rotations be determined from microstructural data (Ghosh and Ramberg, 1978). Let us consider an inelastic matrix subjected to pure shear or homogeneous simple shear. The presence of rigid inclusions in the ductile matrix gives rise to strain perturbations, i.e. local fields of heterogeneous strain, which generally lead to solid-body rotations (Ramsay 1969, p. 52). Only if a perturbation has a symmetrical strain pattern (Fig. 3), will the inclusions retain their initial orientation. This is not the case for progressive simple shear, which invariably rotates all rigid spherical inclusions (Fig. 3B). Because the absolute rotation is chiefly dependent on the strain pe~urbation rather than on the overall simple shear of magnitude y, we find that, for a linear viscous fluid, the solid-body rotation of the unperturbed matrix (4,) differs from the rigid-body rotation of the small spheres ($,). According to Nadai (1950), Jeffery (1923), Rosenfeld (1970) and Ghosh and Ramberg (1976, in press), 4% = tan -’ (--r/2) whereas cpS= -y/2. Note that &, < b/21, but I&I can approach infinity** (no limit as to the number of revolutions). Nevertheless, *According to widespread usage (Nye, 1957; Ramsay, 196’7, p, 124), the antisymmetrical matrix is defined in such a way that all leading-diagonal terms are zero. This convention has not been followed in recent papers (for example, Cobbold, 1977, p. T2). **Dixon (1976, p. 102) put #m = q+, and apparently did not recognize that the two rotations can be vastly different.
T19
Fig. 3. Deformation 1975). Distortion of simple shear (B). The strain, at the onset of
pattern around rigid circular inclusions (modified from Ghosh, initially orthogonal grids due to pure shear (A) and homogeneous original grid lines were parallel to the principal directions of overall progressive deformation.
can be calculated from &, which must be determined from microstructural evidence (Rosenfeld, 1970; Dixon, 1976; Ghosh and Ramberg, in press),
Om
CONCLUSIONS
In summary, the following conclusions may be drawn: (1) The shift of the principal lines of finite strain, from their initial orientation to the final orientation, provides the best measure of finite solid-body rotation, either at a point or for a large domain of homogeneous strain. For small deformations, this shift corresponds to the antisymmetric part of the infinitesimal-displacement gradient matrix. (2) Mechanically passive, primary features with structural polarity are probably the best natural gauges of paleo-rotation. (3) Spherical “rigid” inclusions are rolled by simple shearing only, which is but one mechanism of solid-body rotation. (4) The ductility contrast between “rigid” inclusions and rock matrix gives rise to strain perturbations which can lead to solid-body rotations that differ from those of the adjacent unperturbed regions. ACKNOWLEDGEMENT
I would like to thank my colleague Pierre-Yves Robin for a critical review of the original draft.
REFERENCES Biot, M.A., 1965. Mechanics of Incremental Deformations. Wiley, New York, N.Y., 604 pp. Cobbold, P.R., 1977. Compatibility equations and the integration of finite strains in two dimensions. Tectonophysics, 39 : Tl-T6.
T20 Dixon, J.M. 1976. Apparent double rotation of phorphyroblasts during a single progressive deformation. Tectonophysics, 34: 101-115, Efliott, D., 1970. Determination of finite strain and initial shape from deformed elliptical objects. Geol. Sot. Am. Bull., 81: 2221-2236. Ghosh, S.K., 1975. Distortion of planar structures around rigid spherical bodies. Tectonophysics, 28: 185-208. Ghosh, S.K. and Ramberg, H., 1976. Reorientation of inclusions by combination of pure shear and simple shear. Tectonophysics, 34: l-70. Ghosh, SK. and Ramberg, H., 1978. Reversal of spiral direction of inclusion trails in paratectonic porphyrobl~ts. Te~tonophysics, 51: 83-97. Jeffery, G.B., 1923. The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Sot. Lond., 102A: 161-179. Love, A.E.H., 1944. The Mathematics Theory of Elasticity. Dover Publ., New York, N.Y., 4th ed., 643 pp. Nadai, A., 1950. Theory of Flow and Fracture of Solids. McGraw-Hill, New York, N.Y., Vol. I, 2nd ed., 572 p. Nye, J.F., 1957. Physical Properties of Crystals. Oxford University Press, London, 322 Rankly, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, New York, N.Y., 568 pp. Ramsay, J.G., 1969. The measurement of strain and displacement in erogenic belts. In: Time and Place in Qrogeny. Geol. SOC. London, pp. 43-79. Ramsay, J.G., 1976. Displacement and strain. Philos. Trans. R. Sot. Lond., Ser. A, 283: 3-25. Rosenfeld, J.L., 1970. Rotated garnets in metamorphic rocks. Geol. Sot. Am., Spec. Pap., 129: 106 pp. Schwerdtner, W.M., 1976. A principal difficulty of proving crustal shortening in Precambrian shields. Tectonophysics, 30: TlS-23. Schwerdtner, W.M., 1977. Geometric interpretation of regional strain analyses. Tectonophysics, 39: 515-531.