The formation of crenulation cleavage
j. W. COSGROVE
SUMMARY Investigation of a theory of buckling of an mineral fabrics or finely laminated multi- anisotropic material has enabled the micro- layers. Crenulation cleavage planes are gener- folding of mineral fabrics to be more clearly ally formed as a consequence of mineral understood. It is shown that "crenulations" migration which occurs in response to stress (microfolds) are associated with a variety of gradients intimately associated with the devel- buckling instabilities which develop in stressed oping microfolds.
2~LTHOUOH a large variety of cleavages has been described in the literature it is possible to consider just four "end members" of a cleavage tetrahedron (Fig. x) with all other varieties of cleavage as intermediate forms. Each of the four end members, slaty cleavage, fracture cleavage, pressure solution cleavage and crenulation cleavage can be readily recognised in the field. Slaty cleavage is a penetrative cleavage and is the result of the development of a pervasive mineral fabric. Fracture cleavage and pressure solution cleavage are non-pervasive cleavages. In pressure solution cleavage the discrete planes of weakness that constitute the cleavage are the result of localised pressure solution and migration away from thesite of solution of various mineral species, particularly quartz and calcite. Crenulation cleavage is another non-pervasive cleavage which deveIops in finely laminated rocks and mineral fabrics as a result of micro- buckling together with some pressure solution and redistribution of minerals. Gradations between these four basic types of cleavage are commonly found in nature. Often a fracture cleavage is found superposed on a slaty cleavage. A certain type of crenulation cleavage when fully developed becomes almost indistinguishable from a slaty cleavage. Another type of crenulation cleavage found in association with normal kink bands (P1. I f) may become indistinguishable from pressure solution cleavage. The aim of this paper is to investigate the development of the crenulation cleavage "end member"; only a brief discussion of some of the more closely associated intermediate forms of cleavage is given. Crenulation cleavage has been given various names in the literature, e.g. close-joints cleavage (Sorby i857), ausweichungs-Clivage (Hiem x878), strain- slip cleavage (Bonney I886), cataclastic cleavage (Knopf I93t), transposition cleavage (Weiss I949) , slip cleavage, secondary cleavage and false cleavage (Dale x892 , t894, I897, I899)Runzelcfivage (Born I929)Gleitshieferung (Schmidt I932), Umfaltungsclivage (Sander I934) Schubkluftung (Sholtz i93o), zweite Schieferung (Hoeppener i956 ) herringbone cleavage, shear schistosity, shear cleavage (Mead 194o, Wilson 1946), fracture cleavage (Agron 195o), crenulation cleavage (KnUl i96o , Rickard i96i ) and crenulation foliation (Whitten x966 ). This large number of names for crenulation cleavage reflects the wide variety of microstructures encompassed by the term. This paper demonstrates that all varieties of crenulation cleavage are formed by essentially the same process, and Jlgeol. Sot. Lond. vol. x32, I976, pp. t55-t78. I8 Figs, 2 Plates. PHnted in N. Ireland.
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it is unnecessary to postulate several profoundly different mechanisms to account for the many natural types of crenulation cleavage. The formation of crenulation cleavage can for convenience be divided into two stages. The first stage is the development of buckling instabilities in an anisotropic medium which may be either a mineral fabric (P1. z a, f) or a multilayer (P1. x c-d). The second stage is the development of discrete crenulation cleavage planes which are found closely associated with the microbuckles. This generally involves the processes of pressure solution and mineral redistribution. Until recently the buckling behaviour of mineral fabrics was not under- stood. Conventional buckling theories were unsuitable as they considered either single or multilayer models with various competence contrasts between the layer and matrix or between adjacent layers. None of these models approxi- mated well enough to a mineral fabric to allow them to be used to predict the buckling behaviour successfully. However in the last decade a theory of buckling has been developed which considers the buckling behaviour of an anisotropic material, i.e. a material in which the resistance to shear differs from the resis- trance to compression in the same direction. (Biot i964, Cobbold et al. i971 and Cosgrove i972 ) . Many mineral fabrics approximate well to this theoretical anisotropic material and the buckling theory can be successfully used to account for their buckling behaviour. This theory is summarised below, emphasising those aspects that have particular relevance to the understanding of crenulation cleavage. Although the buckling theory of an anisotropic material can be used to explain the initiation of buckling instabilities that occur in many mineral fabrics and multilayers, it cannot account for the formation of the discrete planes of weakness that are often associated with the crenulations and which constitute the actual crenulation cleavage planes. These planes of weakness are generally the result of pressure solution and mineral redistributions. The intimate association of micro- folding and mineral redistribution (metamorphic differentiation) can be explained in terms of mineral migration in response to stress gradients that are established during microfolding. The nature of these stress gradients and the effect of the detailed features of a mineral fabric or multilayer on the final form of the crenu- lation and crenulation cleavage planes, are discussed in the second part of this paper. The buckling behaviour of anisotropic materials Several examples of microstructures which fall under the general heading of crenulation cleavage are shown in P1. xa-f. It is not difficult to see why at least
n Cleavage
rTessure Solution Cleavage Fio. I. The cleavage tetrahedron.
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two schools of thought arose on the origin of crenulation cleavage. One con- siders that crenulation cleavage is the result of "microfaulting" (P1. If) and the other that it is the result of "microfolding" (P1. ta). In addition to these two types another type has been frequently described: crenulation cleavage occurring in conjugate sets. The first account of conjugate crenulation cleavage was given by Muff (in Peach et al t 9o9). Muff's original diagram and a slight modification of this by Wilson (I96I) are shown in Fig. 2. It can be seen that both writers con- sidered the cleavage to be a consequence of microfaulting. It will now be dem- onstrated that these three varieties of crenulation cleavage are simply different expressions of buckling instabilities developed in anisotropic materials. For the convenience of the following analysis the coordinate directions x and y are chosen to coincide with the symmetry axes of the material. It is assumed that (a) the material is linearly elastic or Newtonian viscous, (b) the material is anisotropic and homogeneous, (c) the material deforms by plane swain, (d) there is no volume change during the deformation and (e) that the maximum principal compressive stress is applied parallel to the coordinate directions (i.e. mineral fabric). From the assumption (e) one may infer that the combined symmetry of the material properties and the stress field is orthotropic, i.e. that the principal stress axes coincide with or are symmetrically oriented with respect to the symmetry axes of the material properties. It is also assumed that the material is in a state of initial stress and that this initial stress gives rise to an initial uniform state of homogeneous flattening within the material. The aim of the analysis is to in- vestigate the types of instability which disrupt this initial uniform state. These instabilities result in ordered, non-uniform displacement patterns which are described by displacement vectors U and V in the x and y directions respectively. By stating the conditions (a) to (e) mathematically and by combining the resulting equations, it is possible to develop a single equation, solutions to which automatically satisfy these five conditions. The buckling equation that results from the analysis is
( Q- ( Q+ (I)
\ b :..
Fxo. 2. (a) Formation of conjugate crenulation cleavage (after Muff in Peach et al. t9o9) (b) A modification of Fig. 2a by Wilson (t96x).
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where 4 is a displacement function related to the displacements U and V
u = & (2)
V - ~4~ (3) P is the differential stress (al -- a3) Q is a measure of resistance to shear and N is a measure of the resistance to com- pression in the direction of the maximum principal compressive stress al. The resistance to shear and compression are stress dependent and it is convenient to express the shear and compressive moduli M and L for any stress condition in terms of Q and N and the stress value. These moduli can be written P M : N +- (4) 4 P r~ = Q +- (5) 2 The ratio M]L is a measure of the anisotropy of a material. Materials for which M and L are very different have a high degree of anisotropy or, simply, a high anisotropy. Equation (x) relates the material (defined by the constants N and Q), the stress state (P) and the deformation expressed by the displacement function 4. If we define the material and the stress state, equation i is reduced to an equation with one unknown, 4, anY displacement pattern represented by 4 which satisfies Equation (x), is a valid deformation pattern for the material. The general solu- tion of homogeneous partial differential equations such as Equation (x) (Green i96x ) is 4 =f,(x + ~y) +f,(x -- ~,y) +f,(x + ~,y) + (x -- ~,y) (6) where ~x and ~ are arbitrary constants, andfx, f~,fs andf~ are arbitrary func- tions. In order to obtain specific solutions (displacement patterns) of the general solution, equation 6, boundary conditions must be introduced to define the arbitrary constants. Biot (1956) chose boundary conditions such that no de- flection or shear stress occurred at the boundaries, consequently buckling dis- turbances that develop when the material is compressed are restricted to the central region of the material. For this reason they are termed "internal in- stabilities". However it is important to remember that the development of buckling instabilitiesin a homogeneous anisotropic material is not restricted to these particular boundary conditions. Theoretical analysis, field evidence and experimental studies show that such materials may develop buckling instabili- ties under a wide variety of boundary conditions. Buckling instabilitiesthat have developed in anisotropic materials where deflections have occurred at the boun- dames are shown in Fig. 3.
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It can be shown that there are at least three types of solutions, (i.e. types of displacement function ~) to Equation (I) which are of geological interest. For convenience these solutions will be termed Type I, 2 and 3. The type of buckling pattern that develops in an anisotropic material depends on the degree of anisot- ropy, a measure of which (v.s.) is given by the ratio of the two moduli M and L. The deformation pattern associated with Type I solution develops in materials where ~M > L (7) when the stress conditions are e > L (8) The deformation pattern associated with Type 2 solution develops in materials where 2M < L (9) when the stress conditions are such that
4 TM(L_M)
~
FIo. 3. Buckling instabilities developed in anisotropic materials where deflections have occurred at the boundaries. (a) An experimentallydeformed phyllite (Paterson & Weiss I966) (b) Buckled anisotropic layer (Johnson i969)
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(If II
ill
Y III II b
FIo. 4. The effect of the displacement pattern represented by equation t t on passive marker lines (a) paralld with, (b) at 45 ° to, and (c) normal to the principal compression. Compare Fig. 4 a,b with Pl. la,b.
is valid when P > L in materials where 2M > L is = -½C{cos i (x - ~y) + cos I (x + ~y)} or ¢ - -C cos I x. cos ~Iy (I I) where I, ~ and C are arbitrary constants. The displacement function ~b, equation I I, defines a particular deformation pattern. The displacements U and V in the x and y directions respectively can be obtained from Equation I I using equations 2 and 3. The effect of these displacements on passive marker lines originally parallel and normal to the principal compression direction is shown in Fig. 4 a and 4 c respectively. From these figures it can be seen that periodic displacements (~ and 2f) occur in both the x and y directions. However, in a mineral fabric in which the maximum compression acts parallel to the fabric, the displacements in the y direction will be more apparent (Fig. 4a). It has been shown (Blot I965) that the buckling resistance of an anisotropic material is dependent on the ratio of the wavelengths (~t,]2~) and is a minimum when the ratio is infinitely small, i.e. when the wavelength in the x direction is infinitely small. In a rock fabric (as opposed to the theoretical material which is homogeneous) the wavelength
PLATE I. (on fadng page) (a) Crenulations and an embryonic crenulation cleavage in a quartz- muscovite schist from the Lukmanier pass Switzerland. (b) A buckled quartz rich multilayer. Treaddur Bay, Anglesey. (c) Crenulation cleavage in a buckled multilayer from the Precambriaax phyUites of Rhoscolyn, Anglesey. (d) Pressure solution planes (x) associated with the buckling of quartz rich layers in the Quartenschiefer, Brigels, Switzerland. (e) Crenulation cleavage developing in association with multilayer buckles. Rhoscolyn, Anglcscy. (f) Crcnulation cleavage in the Devonian slates of the Moscl vaUcy, Berncastle, Germany.
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"*'"'" U j/77m
e. ,,,2cms " f • , lmm
PLATE I.
160
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a
b C
d lmm e 7mm f 2mm
g .2mm_.....__ h 8mm
PLATE 2.
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2~ is not infinitely small. It will tend to a dimension which will ultimately de- pend on the size of the individual mineral grains making up the fabric. In an extensive study of microfolds developed in mineral fabrics the author has observed a direct relationship between the wavelength of the microfolds and the size of the lepidoblasfic or nematoblastic minerals making up the fabric. The buckling displacement pattern (Equation 1 I, Fig. 4 a) is only valid for the first increment of buckling deformation. However, by using finite element techniques and physical models constructed from plasficine to investigate the growth of buckling instabilities into finite structures, it is possible to demonstrate that the geometric amplification of the initial buckling displacement pattern gives a reasonable approximation to the geometry of the finite structures (Fig 4 a and P1. Ia, Fig. 4b and P1. xb). One of the assumptions made in the buckling analysis of anisotropic materials outlined above is that the combined symmetry of the material properties and the applied stress field should be orthotropic. This condition is satisfied when the maximum compression is parallel with, at 45° , or normal to the mineral fabric. It can be argued therefore, that the displacement pattern (Fig. 4 a) would also develop if the maximum compression acted at 45° or 9 °0 to the mineral fabric, provided that in the direction of maximum compression, 2M > L. The effect of the displacement pattern on mineral fabrics in these two orientations is shown in Fig. 4 b,c. Experimental work has shown that the development of buckling instabilities in anisotropic materials is not restricted to conditions of orthotropic symmetry. For example, P1. 2a-c shows three stages in the buckling of a plasticine multilayer in which the principal compression was initially at 20° to the layering. The model multilayer is made up of identical plasticine layers separated by a thin film of vaseline, and confined between two rigid plates which, in accordance with the boundary conditions used in the theoretical analysis, prevent deflections occurring at the boundaries. The asymmetric buckles that develop invite immediate comparison with the asymmetric buckles produced by the theoretical displacement pattern (Fig. 4 a) when superposed on layering at 45 ° to the maximum compression direction (Fig. 4b).
PL~,a'Z'. 2. (facing page) Plate 2 a, b, & c. The initiation and development of asymmetric folds in a model composed of soft plasticine layers separated by a thin film of vaseline. The layering was initially at 20° to the maximum principal compression direction, consequently the combined symmetry of the multilayer and the stress field was not orthotropic. Photomicrographs of crenulation cleavage in the Devonian slates of the Mosel valley, Germany (d), the Precambrian phyllites of Rhoscolyn, Anglesey (e) and the metasediments of the Luk- manier pass, Switzerland (f). Photomicrograph of a slate from Tor Cross, S.E. Devon (g). Cleavage in the Devonian slates of Tor Cross, S.E. Devon (h).
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As we shall see (e.g. PI. Ia,b), a large variety ofmicrofolds (and microstructures) often found in association with crenulation cleavage are the result of Type I instabilities developing in mineral fabrics and multilayers oriented at various angles to the maximum compression. Type 2 solution. The Type 2 solution to the buckling equation (Equation (i)) is valid for materials where 2M < L and under stress conditions such that 4(M]L) (L - M) < P < L. Associated with Type 2 solutions are Type 2 structures. These structures include both normal and reverse kink bands, and differ from Type I structures in that their axial planes are in general not normal to the maximum principal compression. The angle that the axial plane of a Type 2 structure makes with the principal compression direction depends on the anisot- ropy of the material. It ranges from 45° to 9 o°. In a material with a high degree of anisotropy, i.e. one in which M and L values are very different (M[L ~ o) the Type 2 structures make an angle approaching 45 ° to the principal compression direction. In materials with a lower degree of anisotropy the angle between the axial plane of the Type 2 structures and the compression direction increases until in materials with M]L = ½ the axial plane of the structures that develop are at 9 °0 to the compression direction. The inequalities 7 and 9 show that Type I structures develop in materials where M/L > ½ and Type 2 structures in materials where M]L < ½. It is interesting to note that for a material in which M[L = ½, it is irrelevant whether a Type I or Type 2 solution is used to predict the orientation of the axial plane of the structure that will develop. For this material both solutions indicate that structures will form with axial planes at 9 °0 to the maximum compression direc- tion. The relationship between the anisotropy of a material and the angle between the principal compression direction and the axial plane of the structure that develops as a consequence or "buckling" is shown in Fig. 5-
90"
, !
' 0.5 0.0 Modulus Ratio ~ (Anisotrooy) FIO. 5. The relationship between the anlsotropy (M/L) of a material and the angle (0) between the maximum principal compression direction and the axial plane of the structure that develops as a consequence of "buckling".
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In general it can be stated that Type t structures (i.e. structure with axial planes normal to trl) develop in materials with a relatively low degree of anisot- ropy (M and L values not very different) and Type 2 structure (i.e. structures with axial planes oblique to trl), develop in materials with a high degree ofanisot- ropy. It can be seen that the expression of buckling instabilities in anisotropic material depends on both the degree ofanisotropy (Fig. 5) and the orientation of the fabric or layering with respect to the maximum compression direction (Fig. 4, P1. I a, b). Fig. 6a-i shows some of the possible modes of expression of buckling instabilities in materials with different anisotropy and at different angles to the maximum compression direction. Note that when ¢rt is normal to the mineral fabric or layering, as the degree of anisotropy is increased, a transition occurs between internal pinch and swell structures and conjugate "normal" kink bands which form at angles >45 ° to the principal compression direction (Fig. 6 g,h,i). The significance of these conjugate normal kink bands in connection with the formation of conjugate crenulation cleavage, is discussed in the following section. The final appearance of the structures shown in Fig. 6 will depend on several factors including the P.T. conditions that existed during deformation (these will
L OW ANISOTROPY (hf,/'L) HIGH
O G
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IIIll il ,.. Illl IIIIl ~.'.' r \~ ,. Pi" '" f(~l
".. p)} I)) --- t//I ,,t '"' IIIII 26 g i I Fxo. 6. Some possible modes of expression of "buckling" instabilities in materials with different anisotropy (M/L) and at different angles (0) to the maximum com- pression direction.
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determine the dominant mechanism of mineral deformation, e.g. recrystallisation or pressure solution) and the actual nature of the anisotropic material, e.g. if two materials have approximately the same anisotropy but one is a mineral fabric and the other made up of discrete layers, then the final appearance of the buckling structures may be very different. The theory of buckling of anisotropic materials described above has been used to explain the development of a large variety of microstructures (Fig. 6) that are often observed in nature to be associated with crenulation cleavage. The three apparently different types of crenulation cleavage shown in P1. I a,f and Fig. 7 can be seen to be the result of "buckling" instabilities developed in anisotropie materials. The development of Type I structures in a mineral fabric produces crenulations and crenulation cleavage of the type shown in PI. I a. The develop- ment of Type 2 structures in a mineral fabric can produce crenulation cleavages exhibiting the form shown in P1. If and Fig. 7. It has been demonstrated experimentally (Fig. 3, P1. 2a-c) that the develop- ment of the suite of structures shown in Fig. 6 is not restricted to the particular boundary conditions chosen for the analysis nor to the condition of orthotropie symmetry between the stress and the material property. Conjugate crenulation cleavage Conjugate crenulation cleavage was first recorded by Muff (in Peach et al. I9o9) in the Dalradian phyllites of Craignish, Argyllshire. He concluded that the conjugate cleavage planes had the same significance as conjugate faults. The relationship between the conjugate crenulation cleavage and the slaty cleavage in the Craignish area is summarised in Fig. 7. It can be seen that the
' I~ .L b
20 -- 124 ~, 110,° 115~, 119,° 99. 100. 1082 94. 96. 83, FIG. 7. a. Conjugate crenulation cleavage from Craignish, Argyllshire. b. The relationship between the conjugate crenulation cleavage (heavy lines) and slaty cleavage (light lines) (after Knill I957).
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crenulation cleavages are symmetric about a slaty cleavage and were assumed to be the result of the same stress that caused the slaty cleavage. If the crenula- tion cleavages were the result of "brittle" shear failure, then the angle between the conjugate cleavage planes (20, Fig. 7) should be less than 9 °0 . In almost all the examples measured, the angle 20 exceeded 9 o°. Knill (1957) suggested that this angle was originally less than 9 o° and that subsequent flattening had in- creased the angle to more than 9 o°. An alternative interpretation of this conju- gate crenulation cleavage, based on the buckling behaviour of anisotropic materials, can now be presented. It has been demonstrated that conjugate instabilities (Type 2) can develop in materials with a high degree of anisotropy ff the maximum principal compression is either parallel or normal to the fabric (Figure 6 bchi). The angle 20 between the conjugate structures, e.g. Fig. 6h, is equal to or greater than 9 o°. It is suggested that the conjugate crenulation cleavage of the Craignish area is the result of the formation of "normal" kink- like structures (e.g. Fig. 6h) and not of the subsequent flattening of conjugate faults. During the formation of slaty cleavage the rock fabric that constitutes the slaty cleavage becomes more and more pronounced (i.e. the anisotropy of the slaty cleavage fabric increases). It is likely that at some stage in its development (when the shear modulus L in the direction of the maximum compression, is equal to the stress difference P = (tr t -- o'8), see inequalities 8 and io) the slaty cleavage fabric will become unstable with respect to the maximum principal compressive stress. Internal pinch and swell structures may develop or, if the anisotropy of the slaty cleavage becomes high enough, Type 2 structures may develop. These will be conjugate "normal" kink-like structures formed at an angle >45 ° to the principal compression. It follows that as a slaty cleavage develops, instabilities become more and more likely to form, which will tend to inhibit the development of a perfectly uniform slaty cleavage fabric. This may explain why early workers, by examination of thin sections, found considerable difficulty in determining whether slaty cleavage was the result of a mineral fabric which developed at right angles to the maximum compression direction as a result of grain rotation and recrystallisation, or the result of failure along conjugate shear planes which, due to subsequent flattening, were rotated into orientations approximately normal to the principal compression. Conjugate crenulation cleavage may also occur as a consequence of conjugate reverse kink bands (Fig. 6c) developing in a mineral fabric or finely layered material, when the maximum principal compression acts parallel to the fabric. Unlike reverse kink bands, where the layering or fabric within the kink band is not required to thin until quite late in the development of the structure (Fig. 8 a-c), in the development of a normal kink band the layering or fabric within the kink is required to thin at all stages in its development (Fig. 8 d-f). This differ- ence between the two structures may help to explain why normal kink bands occur less frequently than reverse kink bands. The thinning of the layering or fabric within the kink band can be achieved by extension parallel to the layering or fabric within the kink band or by removal of material. The formation of normal kink bands in mineral fabrics is often facilitated by
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the removal of material from the kink band by pressure solution and mineral migration, processes discussed below. This leaves the kink bands relatively rich in less soluble minerals, e.g. mica and hornblende. Consequently these normal kink bands become planes of relative weakness and constitute a crenulation cleavage (P1. If).
The formation of crenulation cleavage planes In order to explain the formation of crenulation cleavage it is necessary to under- stand (a) how various types of instabilities are initiated in stressed anisotropic materials and (b) how these instabilities develop into finite structures. Although the geometric amplification of buckling displacement patterns (Fig. 4a,b) gives rise to structures very like those associated with many varieties of crenulation cleavage, only when fractures occur at fold hinges does this ampli- fication alone produce the discrete planes of weakness that constitute the crenu- lation cleavage. For the formation of crenulation cleavage planes (as opposed simply to crenulations), a redistribution of minerals is usually necessary, at least in the immediate vicinity of the crenulation cleavage planes. However, it is observed that the formation of these cleavage planes is often associated with a systematic redistribution of minerals about the whole of the microfolds. In general crenulafion cleavage is the result of pressure solution, mineral reactions and migrations. These processes are often grouped together under the general heading of metamorphic differentiation. It is possible, however, for a crenulation cleavage to develop simply as a result of microfolding and without the occurrence of pressure solution and mineral migration. Consider the development of microfolds in a rock fabric made up predominantly of platy or acicular minerals (Fig. 9). The orientation of the mineral flakes is approximately constant along any line drawn parallel to the axial trace of the crenulations. As the folds develop and become tighter, the mineral flakes on the limbs become aligned so that they all lie approximately on one plane, producing a plane of weakness (Fig. 9c). Such planes of weakness occur at regular intervals (i.e. on each limb) and constitute a crenulation cleavage which is the result of the mechanical rotation of minerals during folding.
C/
| l-x d[ "t
FIO. 8. Stages in the development of reverse kink bands (a, b, c) and normal kink bands (d, e, f) (h > t = ta > ta) (t > q > t~ > ta).
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Similarly, structures looking remarkably like crenulation cleavage can result from the development of pressure solution planes in a mineral fabric without the development of buckling instabilities. Durney (~97~), for example has described and discussed the generation of pressure solution planes in a variety of rock types including limestone and shales (Fig. ~o). The examples of crenulation cleavage illustrated in P1. z have been produced by the association of microfolding and metamorphic differentiation and can be divided into two groups. The first group, which includes the crenulation cleavages shown in P1. t a,f, is the result of metamorphic differentiation associated with the microfolding of a mineral fabric. The second group, Plate ~b-e is the result of metamorphic differentiation associated with the microfolding of a multilayer.
(A) METAMORPHIC DIFFERENTIATION Metamorphic differentiationin low grade metamorphic environments generally involves three processes: pressure solution, mineral migration and rcdcposition. These processes arc governed by the laws of thermodynamics which indicate the direction in which a reaction or process will tend to move under a particular set of physical conditions. We wish to know the direction of these processes under various pressure and temperature conditions that exist around a developing fold. The thermodynamic potential which determines the direction of a reaction at a particular pressure (P) and temperature (T) is the Gibbs Free energy, G. For a reaction occurring at constant pressure and temperature to bc spontaneous, dG Fxo. 9. Buckling of a micaceous fabric producing regularly spaced planes of weakness due to the alignment of mica flakes on the limbs. I moval of quartz collap of mica frame.work FIo. x o. Removal of material from a locality by pressure solution giving rise to a structure resembling a crenulation cleavage. Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 x68 J. W. Cosgrove defined by ta~ = P. T.n~ where n~ is the number of moles of component i. The larger the chemical po- tential, the less stable the component and the greater its tendency to react chemi- cally or undergo pressure solution. If a phase contains only one component and if the temperature remains constant it follows from the Gibbs-Duhem relation (Turner & Verhoogen 196o ) that d# = VdP (12) where V is the volume. This relationship indicates that increase in hydrostatic stress (pressure) will increase the chemical potential of a mineral. However, in many geological situations, particularly situations in which pressure solution and mineral migration are likely to occur, the stress will be non-hydrostatic. The application of non-hydrostatic thermodynamics to geolo- gical processes including pressure solution has been considered by McLellan (1966) and Paterson (1973) and the conditions of equilibrium between a stressed solid and its solution is found to be /~1 =/~. -- T. S. + 0.V. (13) where /~,, S, and V, are the chemical potential, entropy and volume of the solid respectively. This equation shows that the chemical potential of a solid in a surrounding solution/~1 is directly related to the normal stress 0, acting on its surface. Equations 12 and 13 show that the chemical potential of a mineral and there- fore its tendency to undergo pressure solution will be directly related to the magnitude of the applied stress. In considering pressure solution and mineral migration around microstructures, it will be assumed that the temperature remains constant and that pressure solution and migration occur only in response to stress gradients. For conven- ience we will assume that a "phase" exists along grain boundaries and that mineral species can leave their parent crystal, enter the grain boundary phase, migrate through it and precipitate out onto some other crystal or some other part of the parent crystal. Because the chemical potential of a mineral depends on the state of stress (equations 12, I3), there will be a gradient in chemical potential parallel to any stress gradient. The chemical potential at any point in a crystal can be considered as a measure of the tendency of that part of the mineral to leave the parent mineral and move into the grain boundary "phase". It follows therefore that there will be a concentration gradient established in the grain boundary phase which parallels the chemical potential gradient. Mineral species will migrate through the grain boundary phase as a consequence of this concentration gradient. During the deformation of a rock, stress gradients may be established on var- ious scales, for example stress gradients may be established around individual grains or, on a larger scale, between the hinge and limb areas of a fold. In order Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 Formation of crenulation cleavage 169 to explain the development of stress gradients around the folds in a buckling mineral fabric it is necessary to consider the stress distribution within a stressed multilayer made up of alternating layers a and b, which may be either elastic or viscous, with rheological moduli R, and R~ respectively (Fig. I I). Consider the model to be made up of linear elastic materials a and b. The constitutive equations for layers a and b will be at -- as = R,e, and at -- a3 = Rbeb, where e, and e~ are the elastic strains in the layers a and b respectively. If the multilayer is shortened uniformly, e, = eb. From Fig. I I it can be seen that a3 is the same in layers a and b. It follows therefore from the constitutive equations that the respective magnitudes of at in layers a and b are different. The layers with the higher valued modulus support a larger proportion of the compressive force parallel to the layering than the layers with the smaller modulus. The less "competent" layers are "protected" by the more "competent" layers. This effect can be used to explain how stress gradients develop in certain minerals and layers during folding. (B) STRESS GRADIENTS AND METAMORPHIC DIFFERENTIATION ASSOCIATED WITH THE MICROFOLDING OF A MINERAL FABRIC (i) Metamorphic differentiation in a quartz/muscovite fabric. Microfolding of a planar quartz/muscovite mineral fabric accompanied by pressure solution and mineral migration in the Preeambrian phyllites of Rhoscolyn, Anglesey, has resulted in metamorphic differentiation (Fig. 12). The limbs of the folds are now almost IOO% mica and the hinges consist predominantly of quartz. Using the buckling theory for anisotropic materials given earlier and the arguments regarding stress distribution in a multilayer, presented above, an attempt will now be made to account for this differentiation. Because of the ease with which quartz deforms by pressure solution in low grade metamorphic environments (Cosgrove i972 , Kerrich i975) it can be assumed that during the folding of a quartz/mica fabric, the mica offers a larger resistance to compression than quartz. It follows from the discussion of the stress distribution within a multilayer Fig. I I that al in the mica flakes would be greater than Ol in the quartz. One may also assume that the stress difference al -- a3 increased to some critical value so that the mineral fabric, being aniso- tropic, became unstable and developed buckling instabilities (e.g. Type I buckling instabilities, Fig. 4a). At the onset of buckling, the micas began to rotate away from parallelism to the x direction, Fig. I2. Initially, the maximum principal stress trajectories continued to act along the mica flakes, (the orientation of stress FIG. Ix. (a) For any particu- larly strain e (or strain rate) the L ...... ,L-a / effective stress P (1) -----a I -- as) in the two materials a & b, is different. (b) Schematic stress/strain (or strain !...... :J b rate) plot for two elastic (or viscous) materials a & b making up the I l e multilayer. 0 Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 x7o J. W. Cosgrove trajectories during folding is well illustrated in the finite element work of Die- terich & Carter ~969). However, as the folds developed and the mica flakes on the limbs continued to rotate away from the x direction, the trajectories began to cut "across" the fabric, Fig. z2, and acted once more approximately parallel to the x direction. As folding proceeded beyond this stage (Fig. z2c) the quartz in the limb areas became subjected to a progressively larger compressive stress as it was no longer "protected" by the mica framework (cf. Fig. i2a and c). The magnitude of the stress on the quartz in the limb will increase with the dip of the limb, reaching a maximum when the folds are isoclinal. Because the micas in the hinge area have not rotated, the quartz in the hinge area is still protected. It follows that the stress acting on the quartz in the limbs is greater than the stress acting on the quartz in the hinge. A stress gradient will therefore exist between the quartz in the hinge and the limb and the magnitude of this gradient will increase as the folds develop. This stress gradient will tend to cause migra- tion of the quartz from the limb towards the hinge. However, the stress gradient established in the micas during the development of the folds shown in Fig. ~2, will act in the opposite direction to the gradient established in the quartz. The mica in the hinge areas supports most of the compression and is under a relatively large compressive stress compared to the mica in the limbs (Fig. x2c). Pressure solution of mica will therefore occur more readily in the hinge areas and there will be a tendency for it to migrate from the hinge areas towards the limbs. (ii) Metamorphic differentiation in a chlorite~muscovitefabric. Limbs of microfolds in a chlorite/muscovite rock from the Precambrian of Rhoscolyn, Anglesey (Fig. 13 c) have been completely recrystallised within very well defined zones. X-ray diffraction analysis of the limb and hinge areas confirms that the rock is made up almost entirely of muscovite and chlorite and that the relative proportions of these two minerals in the limb and hinge areas are very different, (Fig. I3a,b. ) The ratio of muscovite to chlorite in the limbs is c. 4:I and in the hinge c. 1:2"5. Observations in areas where the mineral fabric is unfolded show that originally the proportion of muscovite to chlorite was approximately constant. I ~ m:::m ~ ~ C:~l 1~"~,~ ~,,.~"_'~.l u':"--cm2"v"~v"~'~' DP ~'~''J~'~ '~ '~-,"~'1 C I~---0 ,~ ¢"~"] d _--.. ,,I- principal ~- :_-~ 5._j~- , -"_~ T compressive ~ quartz matrix Y ~ flake YL_ x F xo. ~2. A schematic representation of microfolding and metamorphic differ- entiatlon in a quartz mica fabric. Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 Formation of crenulation cleavage 17 r During the conversion of chlorite to muscovite, water is liberated. Because this water can migrate away from the locality of its liberation (see Gresens (i966) on migration of water towards fold hinges) and because chlorite has a larger volume than muscovite, the conversion of chlorite to muscovite at a locality is a mechanism of producing a local volume reduction. Such a reaction will be facilitated by an increase in compressive stress, whereas the breakdown of musco- vite to chlorite, which involves a volume increase will not. The chlorite will therefore be less capable of supporting stress than the mica and it is suggested that the system chlorite/muscovite is analogous to the quartz/muscovite system (Fig. I2), where the muscovite acts as the more competent material relative to chlorite. Following the arguments used in the discussion of the quartz/muscovite folds of Fig. x2, there will exist a stress gradient in the chlorite which will tend to cause migration of the chlorite from the limbs to the hinges. Because of the complexity of the chlorite structure, it is unlikely that "unit chlorite" molecules go into solution. It is more likely that the chlorite undergoes incongruent solution, i.e. parts of the molecule enter solution, and that these migrate and are deposited at various sites around the crystal. Because the conversion of chlorite to muscovite and water can achieve local volume reduction and requires little alteration of the basic silicate framework, it is probable that as the chlorite begins to recrystaUise in response to the relatively high stress it experiences on the limb, it is converted to muscovite by absorbing free K+ ions and rejecting Mg++ and Fe++ ions. This will result in a concentration gradient in Mg++ and Fe++ ions which will parallel the stress gradient in the chlorite. Consequently these mafic species will tend to migrate towards the hinge. At the hinge zone, muscovite is under a relatively high stress compared to the muscovite on the limbs (cf. Figure x2c). It will therefore tend to recrystallise. In doing so it is probable that the free Mg++ and Fe++ ions and the water, concentrated at the hinge, will be incor- porated into the new lattice, resulting in the formation of chlorite. This will II4USCOVITE IM L!MB. ~-- ~ -525 [~1 ~ HIN-----G[32" C M C M 32 ° FIo. x 3. X-ray diffraction peak patterns (CuK~) (a) from the limb area and (b) from the hinge area of a folded chlorite/muscovite fabric (c). Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 i72 Jr. W. Cosgrove result in the build up of K+ ions in the hinge area, as the K + ions from the musco- vite lattice are not incorporated into the chlorite lattice. The build up of K + ions at the hinge due to the formation of chlorite from muscovite, and the absorption of K+ ions in the limbs due to the formation of chlorite from muscovite would result in a concentration gradient in K+ ions which would tend to cause migration from the hinge to the limb. In this way, migration of a few simple species, Mg ++, Fe ++, K + and H,O, in response to stress gradients and the resulting concentration gradients can account for the "redistribution" of muscovite and chlorite during the development of the microfolds. Although the limb areas are totally recrystallised during microfolding, not all the chlorite has been converted to muscovite. It can be inferred from this that the migration of K+ ions to the limb areas was the rate-determining step. As the K + ions are provided by the conversion of muscovite to chlorite, a process involving a volume increase in an essentially compressive environment, this process is likely to be the rate determinator. C) METAMORPHIC DIFFERENTIATION ASSOCIATED WITH THE FOLDING OF A MULTI-LAYER In the Precambrian phyllites of Rhoscolyn, Anglesey, which are made up of alternating quartz rich and mica rich layers c. ]:in. thick, crenulation cleavage has developed on two scales. On one scale as a consequence of the buckling of the mica fabric of the mica rich layers and on a larger scale as a consequence of the buckling of the multilayer as a whole (PI. I C). Using the arguments presented to explain the metamorphic differentiation that occurs in a quartz/muscovite fabric during folding (Fig. 12), it is suggested that the quartz rich layers in the limb areas within the folded multilayer are under a relatively high compressive stress compared to the quartz rich layers in the hinge areas which are protected by the mica rich layers. The quartz rich layers on the limbs will therefore tend to undergo pressure solution and migrate towards the hinge areas (Fig. 14). Some of the folds that develop in the multi- layer have slightly divergent axial planes (Type 2 structures --P1. Ie). In such folds the limb areas where intense pressure solution of the quartz rich layers has occurred are particularly well defined. They are separated from the hinge areas by a narrow zone where the change in dip of the layering occurs very rapidly. The quartz rich layers in the limb region may be completely removed leaving isolated hinge regions in which the multilayer fabric is still discernible, separated by mica rich bands (P1. xe, Fig. I4e , Fig. 15). The minor folds that have developed around the fold shown in Fig. 15 also demonstrate the effect of layer orientation on the style of structure that develops. Compare the structures that form in the hinge area of the large scale fold with Fig. 4 a and the structures that develop in the limbs area with P1. 2b. Crenulation cleavage may also develop in a multilayer made up of alternating quartz rich and mica rich layers in a manner slightly different from that de- scribed above (Fig. 14, cf. Fig. 16 & PI. I d). At stage Fig. 16c the quartz rich layers physically "collide" and further buckling is inhibited. The multilayer "locks Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 Formation of erenulation cleavage 173 up", and stress concentrations develop at the point contacts resulting in the interpretation of the limbs (Fig. i6d). (The situation is analogous to the inter- pretation by pebbles by pressure solution at point contacts.) This process leads to the development of intense pressure solution on the limbs of the multilayer fold and extreme thinning or complete removal of the quartz rich layers on the limbs. Many of the crenulation cleavages developed in multilayers containing layers of different competences (P1. id) are produced in this way.) However, it is not necessary for the limbs of buckles to collide for intense pressure solution to occur on the limbs (e.g. point X in P1. Id), and the author has observed single layer buckles in the Devonian slates of Tot Cross, South Devon, with well developed pressure solution planes developed on the limb/matrix junctions. These pressure solution planes often extend well into the surrounding matrix. In order to understand the occurrence of a pressure solution plane at ,.5,,, • td'J +,1 t~i f P i~ ...... "" .,,, It,~ ;l~tJ + + ~//~+.+. I'm, 133 I, , iq[;I ! +qlll [iili d e f FIG. x4. Various stages in the transposition of a multilayer made up of alter- mating quartz rich (q) and mica rich (in) layers (Precambrian phyllites of Rhos- colyn, Anglesey). ,1meter, I/+cms, FIO. t5. Transposition of bedding associated with small scale folding in the Cambrian Cabitza slates of the Sa Duchessa area, SW. Sardinia. Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 ~74 J. W. Cosgrove the fold limb/matrix junction and the propagation of this plane into the matrix, it is useful to consider the stress distribution around a relatively rigid particle set in a less rigid matrix. The principal stress trajectories and the variation in mean stress and principal compressive stress around a rigid inclusion set in a less rigid matrix are shown in Fig. 17B,A. The mean stress, maximum principal stress and normal stress are all maximum where the particle/matrix interface are normal to the bulk ax direction, and lowest where the interface is parallel to the bulk aa direction. If pressure solution, mineral migration and redeposition occur as a consequence of stress gradients as suggested earlier, it is to be expected that pressure solution will occur in regions a (dotted) and deposition in regions b, (blank) (Figure I7B ). Minerals will migrate from these areas and it is argued that after pressure solution has been initiated, areas a will be analogous to less rigid particles (or even holes) set in a more rigid matrix. The stress distribution around an elliptical hole is shown in Figure I7C. Marked stress concentrations occur at the ends d and e, hence these will be the sites of most active pressure solution. Pressure solution planes will propagate from these regions in a direction normal to the maximum principal compresssive stress, i.e. parallel to the cr3 stress trajectories (Fig. x7B). The initiation of pressure solution planes at fold limb/matrix contacts and the propagation of these planes into the matrix (P1. I d) can be explained if we con- sider a buckled layer to be analogous to a series of relatively rigid particles set in a less viscous matrix (Fig. I7D ). This process can be used to account for the cleavage developed in many of the slates of north and south Devon (P1. ~g,h). (D) THE DEVELOPMENT OF CRENULATION CLEAVAGE IN A COMPLEX ANISOTROPIC MATERIAL The type of cleavage developed in the "slates" of north and south Devon depends primarily on lithology. The slates are predominantly pelitic but contain numerous siltstone and sandstone layers of various thicknesses. The pelific horizons have a penetrative cleavage on which a rather irregularly spaced system of pressure solution planes, parallel or sub-parallel to the penetrative cleavage, is superposed. [ I [ ! FIo. x 6. Various stages in the development of a crenulation cleavage in a multi- layer made up of quartz rich (light) and mica rich (dark) layers (Quartenschiefer of Brigels, NE. Switzerland.) Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 Formation of ¢renulation cleavage x 75 The silt and sandstone layers are generally buckled and cut by well developed pressure solution planes. These planes are often too closely spaced for all of them to be accounted for by pressure solution at the limb/matrix junction. However, despite their close spacing there seems little doubt that a large number of these solution planes are formed by the mechanism illustrated in Fig. x 7. A schematic representation of the various stages in the development of these pressure solution planes is given in Fig. x8. If the siltstone and sandstone layers are sufficiently far apart for their zones of contact strain not to overlap, each layer will develop its own characteristic wavelength 2 depending on its thickness t, and its competence pl relative to the competence of the matrix /zs according to the single layer buckling equation (r4) (Biot I96I ). 2 -- 2zrt a/Pa Pressure solution will be initiated at the limb/matrix contact in the manner shown in Fig. 17. These planes will propagate into the matrix and eventually m --) FIo. x 7. Principal stress trajectories (B) and contours of maximum principal stress (m.p.s.) and of mean stress (m.s.) (A), around a rigid inclusion set in a less rigid matrix. Fig. I7C shows lines of equal maximum principal compression around art elliptical hole under a compression parallel to its minor axis (after Savin i96x ). Maximum compression occurs at d and e. Figure x7D, a buckled layer acting as a series of rigid particles; pressure solution planes are initiated at the limb matrix contact and propagate into the matrix, parallel to the minimum principal stress trajectories. Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 176 I ' !'J~d3". W. Cosgrove ~ 0_.. c Fxo. x8. Possible stages in the development of pressure solution planes in the slates of south Devon. may cut other quartz rich layers (Fig. 18c). The end result of this process will be a closely spaced system of pressure solution planes cutting both the matrix and the buckled layers. Some of these pressure solution planes will be obviously associated with particular buckles, e.g. a and b of Fig. i8c, and others will not, e.g. d and e. The amount of pressure solution that has occurred along a pressure solution plane is found to vary, e.g. two solution planes may coalesce and accen- tuate each other, x and y, of Fig. I8c. Conclusions An understanding of the buckling behaviour of anisotropic rocks, particularly mineral fabrics, has made it possible to demonstrate that the large variety of microstructures that have been grouped together under the general heading of strain slip cleavage or crenulation cleavage are the result of buckling instabilities developed in anisotropic rocks. Mineral migration in response to stress gradients established during the development of these buckling instabilities often results in the formation of discrete zones or planes of weakness. These planes are intimately associated with the buckling instabilities and constitute a crenulation cleavage. Because the geometry of the structures that develop from the buckling instabili- ties and the form of the crenulation cleavage planes are found to depend on many factors, principally the degree of anisotropy of the rock, the nature of the aniso- tropy (i.e. whether it is due to a mineral fabric or fine alternation of layers), Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021 Formation of crenulation cleavage x77 the orientation of the anisotropy with respect to the principal stress axes, and the degree of development of the structure, it is not surprising that in nature the appear- ance of crenulation cleavage is so varied. It is suggested that the redistribution of minerals around a structure (e.g. a fold or boudin) can be used as an indicator of the stress gradients that existed during folding. ACa~OWLEDO~.mZNT. The author thanks: Dr N. J. Price and Professor J. G. Ramsay for help; Dr Price and Mr M. 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Geol. Palaeont. 25, 235-316. TumoR, F. J. & VERHOOO~N, J. 196O. Igneous and metamorphic petrology. MeGraw-HiU, New York. WF~s, J. x949. Wissahiekon schist at Philadelphia, Pennsylvania. Bull. geol. Soc. Am. 60, x689-726. WHrrTEN, E. H. T. I966. Structural geology offolded rocks. Rand McNally, 6x8. Wmso~, G. I946. The relationship of slaty cleavage and kindred structures to tectonics. Proc. Geol. Ass. 57, 263-3o2. I96x. The tectonic significance of small scale structures, and their importance to the geologist in the field. Annls Soe. gJol. Belg. 84, 423-548. Received 7 February x975; revised typescript received t 5 July x975. JoH~ WmLLA~ COSm~OVE,Geology Department, Imperial College &Science & Technology, Prince Consort Road, London SW7 2BP. Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/132/2/155/4892673/gsjgs.132.2.0155.pdf by guest on 25 September 2021