Finite strain and progressive deformation in models of diapiric structures

@ Elsevier Scientific

89-124 Publishing Company, Amsterdam - Printed in The Netherlands

FINITE

AND

Tectonophysics,

28 (1975)

STRAIN

DIAPIRIC

PROGRESSIVE

DEFORMATION

IN MODELS

OF

STRUCTURES

JOHN M. DIXON Department

(Submitted

of Geological

Sciences,

Queen’s

University,

Kingston,

Ont. (Canada)

November 25, 1974; revised version accepted May 7, 1975)

ABSTRACT Dixon, J.M., 1975. Finite strain and progressive deformation tures. Tectonophysics, 28: 89-124.

in models of diapiric struc-

The distribution of strain within and around gravitationally produced diapiric structures was studied through the use of experimental models which were deformed in a large-capacity centrifuge. A new method of model construction was developed which is equivalent to building the model of initially square l-mm elements. After deformation the elements assume shapes which approximate parallelograms and their finite strains can easily be calculated. If several initially identical models are deformed to different extents, the finite strain states of an element in each of the models define points on the deformation path of that element. The deformation path can be used to make estimates of the nature of the internal fabric which would be expected in the equivalent element of the natural structure. This method was applied to the study of the finite strain in diapiric ridges. The models demonstrate that the highest strain is always in the region above the diapir. Within the diapir initial vertical stretching is followed by vertical flattening. Large portions of the structure can be seen to suffer what would in natural examples be called polyphase deformation, even though all of the deformation was due to a single buoyant overturn of unstable density stratification. The strain patterns within the models support the contention that in salt diapirs the buoyant salt has a lower viscosity than the overlying sediments, but that in mantled gneiss domes the reverse is true.

INTRODUCTION

Mantled gneiss domes (Eskola, 1949) are an important structural feature of the core zones of erogenic belts. They represent one species of a varied family of geological structures all of which are thought to be produced by the spontaneous overturn of inverted density stratifications in the earth’s gravitational field. Such structures include small-scale soft-sediment slump features (Anketell et al., 1970), some mesoscopic structures in sedimentary rocks (Stephansson, 1971), salt anticlines and domes (Balk, 1949; Braunstein and O’Brien, 1968) and batholiths. The size range involved is thus from a few centimeters to a few tens of kilometers. It has also been suggested that the same process may occur on a much larger scale within the earth’s mantle (‘mega-undations’), serving as the engine which drives the wandering litho-

90

spheric plates across the face of the globe and is the ultimate cause of all geodynamic processes (Van Bemmelen, 1972; Ramberg, 1971). A knowledge of the dynamics of the process of buoyant overturn in layered systems is thus of fundamental importance for the understanding of a wide range of geological structures. Geometrical field studies of these structures are important but are of somewhat limited value in that they suffer from limited exposure and our ignorance of the initial geometry of the system. It is therefore instructive to study the process through the use of theoretical and experimental models. Previous investigations Several geologists have used experimental models to study the formation of domes and diapirs, but most of these studies have been aimed at determining the conditions governing the overall geometry of the domes rather than their internal structure. This is because most of the early models were constructed with low-viscosity fluids such as oil and syrup and so could be studied only during deformation. One of the first studies (Escher and Kuenen, 1929) produced folds in models of salt diapirs by forcing a clayparaffin multi-layer through a cylinder, an unrealistic experimental set-up. Nettleton (1934) was the first to demonstrate that superposed viscous fluids with inverted density stratification develop spontaneously into a series of ridges and domes. Following Hubbert’s (1937) classic work on scale-model theory, Dobrin (1941) and Parker and McDowell (1955) did careful model studies applicable to salt-dome tectonics. In the early 1960’s Ramberg (1963, 1967) first used a centrifuge to study the development of gravity-driven structures. The advantage of the centrifuge technique is that relatively stiff materials can be used to construct realistic, complex models which are then deformed by the centrifugal body force which simulates gravity increased by a factor of several thousand. The deformed models can then be dissected and studied at leisure in the laboratory without undergoing further deformation under their own weight. The centrifuge technique has been developed to a high degree of sophistication in more recent works, and complex models simulating orogenesis and continental drift have been produced (Ramberg, 1971, 1972; Ramberg and Sjostrijm, 1973; Stephansson, 1972). Ramberg’s diapir models show very clearly the nature of the deformation within and around the rising bodies, but in general only qualitative measurements of the strain distribution could be derived from the stretching and buckling of the thin competent layers within the models. Theoretical fluid dynamical treatments of the gravitational instability of layered systems have been published by various workers, including Taylor (1950), Danes (1964), Selig (1965), Biot and Ode (1965) and Ramberg (1968a,b). These works permit the dominant wavelength in the unstable system to be determined, as well as its initial rate of growth. Ramberg’s (1968a,b) treatment allows the displacements of points within the layers to

91

be determined, up to the stage where the dome amplitude equals about 10% of the dominant wavelength. However, he has not done an analysis of the strain distribution. Fletcher (1972) published the results of the application of a similar mathematical treatment to the emplacement of gneiss domes and gneiss anticlines, with particular reference to the Oliverian domes of New England. His treatment also provided an indication of the internal strain and stress distributions at a dome amplitude approximately equal to the thickness of the source layer. Berner et al. (1972) also analyzed the growth of several model diapirs by the use of the mathematical finite-element method and compared the results to those from the fluid dynamical and experimental methods. The published results include only the external shape of the diapir and the stress distributions around it. The finite-element method provides all the data needed to compute the finite (and infinitesimal) strain distributions but this has not yet been done. Present investigation This paper describes and experimental model study of some aspects of the formation of mantled gneiss domes. A new method of model construction was devised in order that the finite strain for each of a large number of small elements within and around the dome could be determined at several stages in its growth. The method has been described in detail by the author in a previous paper (Dixon, 1974). The finite strain states of an element at the stages of its deformation define points on the deformation path of that element. Knowledge of the path serves to constrain estimates of the nature of the internal fabric of the corresponding rock element in the natural dome. The resulting fabric estimates provide a guide for the interpretation of natural gneiss domes. The experimental models were deformed in a large-capacity centrifuge. THE MODELS

Two series of experimental models were constructed following the method described by Dixon (1974). The first, consisting of models WD-4, WD-2, WD-8 and WD-3, were initially identical with phsycial properties and geometry as shown in Table I and Fig. 1. The second, consisting of models WD-5, WD-6 and WD-7, had the same initial geometry but had layer properties as listed in Table II. The principal difference between the two series was thus the viscosity contrast between the buoyant and lowest overburden layers. The models of the first series, with stiff buoyant layer, were run in the centrifuge at 2OOOg for the time intervals shown in Table III, in order to produce structures of different amplitudes. The models all produced cylindrical ridges and could thus be analyzed by the method outlined in Dixon (1974). The results and interpretation of the analyses are presented in the present paper.

TABLE

I

Material properties of the layers in models WD-4, WD-2, WD-8 and WD-3 Layer No.

Material

Thickness (mm)

Specific gravity*

Viscosity (poises)

10.0 6.0 (1.0 mm laminae) 6.0 (1.0 mm laminae)

2.01

-105

I_--__ 1

2 3 4 5 6

Plastic-base painter’s putty Dow Corning silicone putty ICI silicone putty Dow Corning silicone putty Dow Corning silicone putty Dow Corning silicone putty

1.40

1.5 . 10”

1.56

3.49 * lo5

4.0

1.50

5.61 1 lo5

4.0

1.48

5.61 * 10”

4.0

1.49

~_-----

5.61 . lo5

-- _I_

* Specific gravity of silicone putty adjusted with -250

mesh btirite powder.

The models of the second series, with soft buoyant layer, failed to develop cylindrical ridges when deformed in the centrifuge. Instead they rapidly formed a central dome with elliptical or circular horizontal section. Consequently, serial vertical sections could not be superimposed and the quantitative strain analysis was impossible. Some interesting qualitative observations concerning the overall strain patterns can be made, however, and are presented at the end of this paper. The strain patterns of the two series are strikingly different.

TABLE

II

Material properties of the layers in models WD-5, WD-6 and WD-7 Layer No.

Material

1

Plastic-base painter’s putty ICI silicone

2 3 4 5 6

Putty

Dow Corning silicone putty Dow Corning silicone putty Dow Corning silicone putty Dow Corning silicone putty

Thickness (mm)

Specific gravity*

Viscosity (poises)

10.0

2.01

-lo5

(!I: mm laminae) 6.0 (1.0 mm laminae) 3.0

1.42

3.49 - lo5

1.56 1.55

1.50. lo6 5.61 ’ lo5

3.0

1.54

5.61 * lo5

3.0

1.52

5.61 * lo5

* Specific gravity of silicone putty adjusted with -250

mesh barite powder.

83 TABLE III Deformation histories of finite-strain models _,“___ ___.~~ _~. ___“.__._ __. ----II Model No.

__WD-4 WD-2 WD-8 WD-3 WD-5 WD-6 WD-7

._-_

Temperature 09 _ ___“._~__ 21 22 25 21 21 22 22

Centripetal acceleration (g) _I..--

--

_-

2000 2000 2000 2000 2600 1000 2000

Models with a relatively stiff buoyant

Time of run (seconds) 310 405 240 505 120 540 180

layer

General statement Figures 2 through 5 illustrate typical se&ions through models WD-4, WD-2, WD-8 and WD-3, respectively, after they were deformed in the centrifuge. In the corresponding composite drawings, Figs. 6 through 9, the deformed shapes of the initially square elements can be seen. The shape of each approximately homogeneously deformed element was measured and its finite-strain parameters were calculated by the method described in Dixon (19’74). As was mentioned in that paper with reference to model WD-4, there was found to be a moderate range of area changes, AX=, for elements in both the region of the diapir and the end of the source layer. This variation was ascribed to errors of tracing and measurement in the prep aration of the composite drawing, As such, the errors can be taken as approximally random. Wowever, an element that has a significant area change apparently deviates from the condition of plane strain and consequently has I

Fig. 1. A schematic drawing showing the initial configuration of the models (left side represents the vertically laminated half, and right side the horizontally laminated half) with material properties as listed in Tables I and II. The triangle of material No. 2 in the center of the model indicates the position of a triangular prism which extends along the length of the mode1 and serves to initiate the growth of a cylindrical ridge. The body force acts downwards during the run in the centrifuge. Fram Dixon (1974).

Fig. 2. Sections through the two halves of model WD-4.

finite-strain parameters (principal extensions, fox instance) that axe not comparable tf3 those elements with different areas, In order to make the finitestrain data for each element comparable by eliminatttingthe random area fluctuations, it was found to be convenient to normalize all the elements to unit area, The normalization was accomplished by multiplying the principal extensions by the inverse of the square root of the &m&t’s area (eq. 1).

95

A

Fig. 3. Sections through the two halves of model WD-2.

where the asterisk (*) indicates the normalized value. The normalization procedure is equivalent to applying a homogeneous dilation to each element without changing its shape. The normalized finite-strain data derived from the four models of a diapiric ridge can be used in several different ways to illustrate the deformations involved in its growth. The finite strain states of the elements in each model represent the ridge at the corresponding stage of growth and the dis-

96

Fig. 4. Sections through the two halves of model WD-8.

tribution of the strain provides a means of interpreting the stage of growth of a natural diapir. The several strain parameters and their distribution in each model are discussed below. The finite strain states of an element in all four models define the deformation path of that element. Knowledge of the deformation path followed by a particular element in the model allows the internal fabric of the corresponding element in the natural structure to be interpreted in terms of its

97

Fig. 5. Sections through the two halves of model WD-3.

strain history. The deformation paths of the elements in the diapiric ridge are presented and discussed at the end of this section.

Finite strain distribution The distribution of the maximum finite extensive strain in each of the four models is illustrated by the contour lines in Figs. 6 through 9. The figures show that at the earliest stage the greatest strain in the buoyant layer

98

Fig. 6. A. The composite drawing of the deformed elements in Model WD-4. B. Contours of the values of the maximum principal extension (1 + el )* in Model WD-4. Light stipple = values less than 1.10. Heavy stipple = values greater than 4.00.

Fig. 7. The same for Model WD-2.

100

is a vertical stretching in the core of the diapir. As the structure grows, the vertical-stretching zone moves downwards and the highest strain becomes a horizontal stretching on top of the dome (Fig. 7). As the diapir reaches the surface and starts to spread laterally this trend continues. In the last stage shown the maximum extension is on the order of 10 : 1 in both of these sites. The form of the diapir at the last stage of growth is strikingly similar to that of the post-erogenic granite intrusions in South Greenland described by Bridgwater et al. (1974). In the soft overburden the greatest strain is at all times a horizontal stretching above the diapir, averaging 4 : 1 at first, but reaching a value as high as 60 : 1 in Fig. 9. Another region of high strain in the overburden develops beside the dome trunk after the hat starts to spread. The shapes of the elements in this region show that the strain begins as a layer-parallel shear (Figs. 6 and 7) but develops into strong (10 : 1) vertical stretching during the spreading of the dome; The areas with principal strain less than 10% are indicated by light stippling. In the central part of the dome core in the first two figures, this region

Fig. 8. The same for Model WD-8.

divides the upper zone of horizontal stretching from the lower zone of vertical extension. It extends out towards the flank of the dome and in model WD-4 extends down the flank as well. Elements in this region have undergone no strain but have been rigidly rotated. As the dome grows the region of low strain becomes smaller and finally vanishes (Fig. 9). There are only very restricted regions of less than 10% strain within the overburden, partly because it has a lower viscosity. Naylor (1968) reported that the Mascoma mantled gneiss dome near Hanover, New Hampshire, had apparently been only slightly deformed. As can be seen from Fig. 6, although the dome has risen to a considerable amplitude, such that its flanks are almost vertical, there is a distinct horizon within it in which the strain is low (everywhere less than 20% and in part less than 10%). It seems likely that this is the horizon sampled by Naylor in the Mascoma dome. Regions of small strain within otherwise highly deformed terranes can be as useful in the interpretation of the region as are the minor structures. The

Mascoma dome is a case in point. Naylor interpreted it to be a doubly plunging anticline rather than a dome because it lacks the small-scale structures associated with domes. On the contrary, it could equally well be interpreted to be a dome because it displays the lack of structures characteristic of a specific horizon within a dome, as illustrated by Fig. 6. It is assumed by most workers that in deformed rocks slaty cleavage and probably schistosity record the orientation of the plane of maximum finite extensive strain. This conclusion is based on both field and experimental studies (see for example the discussion in Ramsay (1967), pages 177-182, and the field and experimental studies of Roberts (1971) and Roberts and StriSmg&rd (1972). In the models under discussion here, this plane is everywhere represented by the direction of maximum finite extension. It is therefore instructive to study the orientations of the lines of maximum finite extensive strain in the light of the assumption that they record the probable orientation of schistosity in the rocks of the natural structure. It should be remembered that the models represent diapiric ridges or gneiss-anticlines, which have a horizontal cylinder axis and form under conditions of plane

Fig. 9. The same for Model WD-3.

strain, rather than circular diapirs which have radial symmetry and form under conditions of radial flow. The consequences of this simplification are discussed below. Figures 10 through 13 show the orientations of the lines of maximum finite extensive strain for models WD-4, WD-2, WD-8 and WD-3, respectively. The lengths of the lines are approximately proportional to the magnitudes of the strains. They are the long axes of the finite-strain ellipses. Also shown are the formerly horizontal element boundaries, which represent a hypothetical primary layering in the rocks. These figures therefore show ‘beddingcleavage’ relationships throughout the domes and their overburden. There is some suggestion that cleavage, or at least slaty cleavage, does not form in rocks at strains of less than 30% (Ramsay, lot. cit.). This figure doubtlessly depends on the nature of the rock involved and so may not be applicable to high-grade gneisses. However, it is interesting to note the distribution of zones with less than 30% strain, which are shown by the contours on Figs. 6 through 9.

104

1

Fig. 10. Orientations + eI )* for elements

and approximate in model WD-4.

relative

magnitudes

of the principal

extensions

(1

Schistosity and bedding are in general parallel only in the source layer away from the dome and in the overburden above it. There is also a restricted zone in the center top of the dome in which this condition also holds, especially at the very high amplitude. Note that on the flanks of the dome, in both the buoyant layer and the overburden, the schistosity is at some angle to the layering in an orientation which is axial-planar to the characteristic reverse drag folds found in natural domes. The schistosity in natural gneiss domes in general fits the pattern presented here, which lends support to the belief that it represents the principal plane of flattening in the rocks. It is interesting, however, that the beddingparallel schistosity found high in the dome appears to require considerable amplitude growth before it occupies a significant volume of the dome. Prior to this stage the principal fabric in the core of the dome records vertical stretching (Fig. lo), and the horizontal foliation is strictly a contact effect. The schistosity pattern in the overburden of natural domes does not appear to fit with the hypothesis that it is a principal plane of flattening. Rosenfeld (1968) recorded very large rotations of garnet po~hyrobl~~ within the schistosity in the rocks overlying the Chester dome of Vermont.

Fig. 11. The same for model WD-2.

The schistosity in these rocks is in general parallel to the compositional layering and contacts. It appears that this schistosity is instead a plane of shearing strain. The orientations of directions of maximum shear strain in each element are illustrated in Figs. 14 and 15 for the overburden layers in models WD-4 and WD-2. The reason for the difference between the mechanical significance of schistosity in the dome and overburden is probably that the overburden rocks, being pelitic schists and mafic gneisses in most natural domes, have a strong planar fabric parallel to the compositional layering prior to dome growth, whereas the buoyant material, acid gneisses in general, is less anisotropic. Furthermore, the nature of the deformation in the overburden due to rise of the dome is predominantly a layer-parallel shear. It is energetically favourable for the rock to deform by slip on the pre-existing foliation. The geometry of this kind of deformation is discussed by Schwerdtner (1973). On the other hand the core rocks, having no strongly preferred slip planes, develop instead a foliation in the local plane of flattening (an example of the

Fig. 12. The same for model WD-8.

formation of a foliation in massive rocks is described by Ramsay and Graham, 1970). The strain data presented in Figs. 6 through 15 can be used to construct maps of bedding-cleavage relationships and strain versus cleavage-orientation relationships. Such maps could be used to interpret data collected from natural domes and would be of assistance in identifying what stage of growth the dome has reached and what level within the dome has been sampled. Progressiue ~e~or~u tion

In studying deformed rocks, the structural geologist attempts to deduce the deformation history of a domain based upon his observations of the fabric of the rock, which represents the present, or final, state of strain. He can make only crude estimates of the initial configuration of the body. He generally has only a poor two-Dimensions sampling of its geometry. Finally, he can but guess at the mechanical significance of the fabric

Introduction.

Fig. 13. The same for model WD-3.

elements. Consequently he is often frustrated in his attempts to unravel the deformational path. The experimental method presented here provides a means of determining the deformation path followed by small domains within an evolving geological structure. Within the constraints of two-dimensional geometry and imposed boundary conditions, the method can be used to model natural geological structures and to provide the structural geologist with an estimate of the deformation paths followed by the rocks in the domains he studies in the field. It should be emphasized that the deformation path can be completely defined only if the strain states are known for all times throughout the deformation. Consequently, the present method, which provides points on the deformation path, does not uniquely define it. However, it does provide an estimate that can be used to test hypothetical paths derived from field study. The internal fabric of a deformed body of rock can be thought of as a filtered record of the strain history of the body. The filters are a complex

108

Fig. 14. Orientations of the maximum finite shear strains for elements in the in mode1 WD-4.

overburden

combination of the rock’s physical and chemical properties as well as the temperature and pressure to which it was subjected. It is therefore in general impossible to predict the fabric of a body of rock solely on the basis of a knowledge of the deformation path. Thus the deformation path cannot be deduced from the naturally observed fabric, and the rock fabric cannot be deduced from the experimentally determined path. However, the two procedures can profitably be used in concert, each to test the predictions based on the other. Hopefully the results of this combined approach will be, first, a better knowledge of the deformation paths followed by small elements within natural structures, and second, the definition of the mechanical significance of fabric elements such as schistosity. Progressive deformation in diapirs. The finite-strain distribution in each of the four initially identical models WD-4, WD-2, WD-8 and WD-3 can be used to define the deformation paths followed by the elements during the growth of the diapiric ridge. In each model the elements are numbered as follows: elements in the buoyant layer have numbers between 1 and 200 and are

109

Fig. 15. The same for model WD-2.

numbered from top to bottom in columns beginning at the crest of the diapir. The elements in the lower overburden are numbered in the same way with numbers beginning at 201. This numbering scheme must be slightly modified in models WD-2 and WD-8 in order to compensate for an inaccuracy introduced during the construction of these models. In models WD-4 and WD-3 the initiating ridge beneath the diapir has a half-width of 15 element columns. However, in model WD-2 its half-width is 17 columns and in WD-8, 12 columns. This error means that there is not a strict correspondence between numbered elements in the diapirs. The error can be corrected in an approximate way by omitting two columns of elements from the numbering in model WD-2 and by interpolating three extra columns in the numbering in WD-8. Thus in the presentation of the deformation paths of the elements below, columns six and ten are omitted from WD-2, and extra elements are interpolated between columns four-five, six-seven and eight-nine in WD-8. The two-dimensional finite strain of an element can be completely defined by specifying the values of four parameters: the two principal extensions,

110

(1 + el)* and (1 + es)*, the orientation of the strain ellipse, l3;, and the rotational component of the strain, $ (for precise definitions, see Dixon, 1974). A further simplification is possible in that if the deformation conserved area the two principal extensions are related: (1 + el)* (1 + e3)* = Area of element = 1.0 Therefore the strain state of an element is completely defined by the three independent parameters (1 + el )*, 19; , and 4. The deformation path of an element is the locus of its strain states throughout the deformation. In the present two-dimensional plane-strain case it can only be correctly defined in three-space, or, more conveniently, in two projections onto orthogonal planes. Suitable plots for displaying the path of an element are (1 + el )* vs. I$ and (1 + e, )* vs. 0; . Figures 16 and 17 show these plots for elements in the buoyant layer. Each point represents the element at a stage in its deformation. The points are joined in the sequence WD-4, WD-2, WD-8 and WD-3. In the plots of (1 + e, )* vs. @the progressive deformation begins at the origin, while in the plots of (1 + e, )* , the direction is shown by the small arrowheads. Of all the elements in the buoyant layer, those in the center of the diapir, numbered 1 through 6, have the most easily interpreted deformation path. Figure 16 shows that the strain for all these elements is irrotational. This is because these elements are located on the plane of symmetry of the structure. Figure 17 provides more information about the nature of their deformation paths. Elements 1 and 2 undergo continuous extension in the horizontal direction, and thus rocks in this region might be expected to exhibit a simple flattening fabric. On the other hand, elements 3 through 6 all begin with a period of vertical stretching. The vertical extension is subsequently reversed: the elements return to their initial configurations and then continue to be extended horizontally. Since the deformation is coaxial, the early plane of flattening is not rotated to a new position. Rather, a new plane of flattening forms perpendicular to the first one. Thus the principal plane of flattening does not always exist: it vanishes (in the mathematical rather than textural sense) when an element returns to its original configuration and reappears in a different orientation, without rotating, when the deformation continues. Natural rocks deformed in this way would be expected to exhibit the two planes of flattening in the form of two perpendicular foliations, the first of which would be crenulated or folded about the second. This type of deformation, in which the material is first extended and then compressed in the direction of flow, has been suggested as the cause of many naturally occurring overprinted fabrics (Price, 1972; Fyson, 1971). The plots of (1 + el )* for elements 3 through 6 have a characteristic rectangular shape. Elements in the region between numbers 10, 12, 40 and 42 in Fig. 17 exhibit a modified form of this shape due to varying amounts of rotation involved in their strains. The deformation paths for all these elements are non-coaxial, and although the magnitude of the principal strain first increases, then decreases, and finally increases again, it never re-

vs.e;

VS.e; ,

111

turns to its initial unit value. This implies that the principal plane of flattening exists throughout the deformation and simply rotates through the element as the deformation progresses. The internal fabric of the natural rock body may be expected to reflect this type of progressive deformation. However, it is to a great extent dependent on the nature of the rock and the mechanisms of deformation which operate within it. For example, if the rock fabric is recorded by minerals which can readily reorient themselves in a rotating stress field there may be only one schistosity, which records the final orientation of the principal plane of flattening. On the other hand, if d certain finite amount of shortening is required for the formation of schistosity there may be several schistosities in different orientations, later ones deforming earlier, each recording a portion of the deformation path. Knowledge of the path therefore provides a basis for discerning fabrics compatible with the path but does not uniquely define a particular fabric. Elements in the top row of the buoyant layer, numbered 13 through 73, exhibit in general small but continuously increasing values of the principal extension through time together with clockwise rotation of the principal extension direction (Fig. 17) and a uniform negative increase in the amount of the rotational strain (Fig. 16) which reaches a maximum value of about -180” for elements 25-67 in model WD-3. These maximum values are found in the strongly overturned hat of the high-amplitude diapir in this model. A large portion of the rotation represents the physical overturning of the element relative to the original horizontal datum rather than rotations of the principal directions of strain relative to the material in the element. These elements have principal extensions rarely greater than 1.75. They might be expected to exhibit a fairly simple flattening fabric in a natural diapir. Elements in the second row, numbering from 32 to 74, have almost constant values of their maximum extensions and undergo the clockwise rotation of their axial positions through time. Their strains are so uniformly low that the fabric within them would probably be poorly developed. Another interesting group of elements consists of numbers 59, 60, 65, 66, 71 and 72. These elements are located, in the final two models, in the narrow stem of the dome. The path plots show that the maximum extensive strain is produced in these elements in model WD-8, while in the final model the strain is reduced. This illustrates again the early extension and subsequent recompression which affects some elements. In model WD-8 these elements are stretching vertically as the dome hat starts to spread, while in WD-3 they too are caught up in the horizontal spreading. The next to last column in Figs. 16 and 17 illustrates the deformation paths for some elements away from the dome. All of these elements exhibit progressively increasing horizontal stretching, and the deformations are all virtually coaxial. They demonstrate that flow toward the diapir continues beyond the rim syncline, notably in elements 104-108. Fabrics in this re-

112

52

-3

B

60

59

56

57

56

L

1

,

i

I

d

s

---

doe

c-m&_

-

IlS

D

f2Q.

119

__-.--

__..__

if7

._-..-_

ff6

-.-_

Fig. 16. Plots showing the deformation path projected onto the (1 + el) * - Q,plane for elements l-34, 103-108 and 11 5-120 in the buoyant layer. The deformation path for each element is plotted in its own numbered square. The points record the finite strain of an element in each model, and are joined by straight line segments in the time-sequence origin: WD-4 : WD-2 : WD-8 : WD-3.

B

e

d M

11s

+ ,&

A

~

-_____

60.. %_66.

sa 6s.

tlb.,IL

c

‘11.

-a

L

m--_ ._ e4 -I

-

-------

x-h______83 -*

--

-

loa

107

10s

104

re---

-

-

D

l!u

119

116

Fig. 17. Plots showing the deformation path projected onto the (1 + el)* - 6; plane for elements l-84, 103-108 and 115-120 in the buoyant layer. The deformation path for each element is plotted in its own numbered square. The points record the finite strain state of an element in each model, and are joined by straight line segments in the time sequence WD-4 : WD-2 : WD-8 : WD-3 as indicated by the small arrow-heads.

-.-

54

53.

%,

,

ac

L

F g

116

(WD-4)

(0)

(WD-2)

(~~-81

(WD-3)

Fig. 18. The five stages in the progressive deformation of element number 33. The external rotations have been removed by plotting the element with its original horizontal restored to the present horizontal. The arrows represent the long axis of the finite-strain ellipse at each stage.

gion would record the strong coaxial vertical flattening which continues throughout the deformation. The final column illustrated records the paths for elements somewhat farther from the rim syncline. Here too the strain is coaxial, but it reaches a maximum value in each element at about the time the rim syncline becomes pronounced, that is, between the second and third models in the series. Most of the elements within the core of the diapir exhibit a continuous clockwise rotation during the growth of the structure. The orientation of the principal extension direction rotates clockwise, and the rotational component of the strain increases negatively throughout the deformation. This is because all the elements illustrated are located in the right-hand half of the structure and are involved in the convection-like overturn of the system. The rotations are measured relative to the present horizontal datum. If they were instead measured relative to the present position of the original horizontal in each element, thereby eliminating the external rotation of the element, some elements would be seen to undergo reversals in the sense of their rotation. For example, element number 33 (see Figs. 10 through 13) undergoes first a positive, followed by a negative and then again positive internal rotation. The five stages in the deformation of this element are shown in Fig. 18. The external rotations have been removed by plotting the

#

Qi (interna\)

(i0terncrl)

-+50*

150°

a-Q 1.2

1.3 U+eJ

o

--5o"

100' _.;;/ 50”

W,> Fig. 19. The deformation path of element number 33. @ (internal) is the internal rotation of the principal directions. 0; (internal) represents the orientation of the principal strain (1 + el)* relative to the base of the element. All angles are positive in the anti-clockwise direction.

117

my\ (4

(b)

/r\

/--7--y (c)

(d)

Fig. 20. The hypothetical fabric within element number 33, shown schematically. In (a) the undeformed element contains a vertical quartz vein which becomes boudinaged during deformation to (b). The sense of shearing strain is reversed in going to (c) and the boudins are sheared and shortened into asymmetric folds. A further reversal of the sense of shear, combined with a component of ‘vertical’ extension, causes increased separation between the folded boudins and horizontal compression of the fold shape (d).

element with its original horizontal restored to the present horizontal. The deformation path is illustrated in Fig. 19, with 0; now representing the orientation of (1 + el)* relative to the base of the element, and 4 the internal rotation of the strain axes. The oscillation of the sense of internal rotation can clearly be seen in this figure. The fabric of an element subjected to such a non-coaxial deformation would be much more complex than might be assumed on the basis of the finite-strain geometry. For example, suitably oriented competent sheets in the rock body, such as originally vertical quartz veins, could exhibit folded boudinage indicating compressional strain along a direction which had previously been extended (see Fig. 20). This type of analysis, in which the path of the deformation is measured in moving internal, as opposed to moving external coordinates (for a discussion of the different types of coordinate frames useful in the study of deformation paths, see Elliott, 1972), can be applied to each element in an attempt to predict the internal fabric and distribution of small-scale structures within the element. The most interesting elements in the overburden layer are those located in the contact zone against the buoyant layer, especially beside and beneath the hat of the diapir where the strain may possibly take place by a simple-shear mechanism on a pre-existing foliation (see above, page 105). Unfortunately these elements are so inhomogeneously deformed in the models that their strains cannot be accurately determined. The other elements in the overburden are approximately homogeneously deformed. However, due to problems of experimental error in the assembly of the models and also the very large strains in the overburden above and around the high-amplitude diapirs in models WD-8 and WD-3, it is difficult to determine the correspondence between equivalent overburden elements in the different stages of growth. Consequently no attempt has been made to plot the deformation paths for overburden elements.

118

Models with a retatively soft buoyant

layer

As was outlined in the introduction to t.his section, a second series of models was constructed in which the buoyant layer had a lower viscosity than the overburden. The purpose of these models was to demonstrate the difference between the strain patterns of systems with different viscosity contrasts. Due to the low viscosity of the buoyant layer these models did not produce cylindrical ridges. They could not, therefore, be analyzed in detail. However, some qualitative conclusions can be drawn from them.

E3

Fig. 21. Sections through the vertically laminated parts of models WD-5, WD-6 and WD-7, in which the buoyant layer has a lower viscosity than the overburden.

Figure 23 illustrates four vertical sections through models WI%5, WD-6 and WD-7. All the sections shown were cut through the portions of the models with initially vertical laminations because they best demonstrate the nature of the deformation within the model layers. The four sections can be thought of as representing progressive stages in the growth of the dome. The largest amount of strain in alI. these models is found in the overburden layer above the high-amplitude diapir. This situation is identical to that found in model WD-3 (Fig. 5), and in both cases is due to the lateral spreading of the dome material when it reaches the top surface of the model. Apart

120

from this similarity the strain patterns in the two model series have very little in common. In the first series of models the overburden layer deformed by what appeared to be layer-p~~lel shear. The vertical l~inations were thrown into arcs convex away from the dome indicating what has been called ‘intrastratal’ flow (Rosenfeld, 1968) with drag against both the underlying buoyant layer and the overlying, slightly more viscous upper overburden. In contrast to this pattern, the overburden in the second series (Fig. 21) exhibits almost no internal strain due to its high viscosity. The entire overburden layer beside the flanks of the dome has been rigidly rotated through 90” and more but has undergone virtually no internal deformation except in the hinge of the rim syncline. The strain within the viscous buoyant layer of the first series of models was in general moderate. It exceeded values of 100% in restricted areas, notably the top of the spreading hat and the stem of the high-amplitude diapir in models WD-8 and WD-3. The moderate nature of the strain can be seen qualitatively from the gentle convex-towards-dome curvature of the vertical laminations in Figs. 2 through 5. In contrast the soft buoyant layer in the second series of models has undergone relatively severe intrastratal flow toward the rising diapir (Fig. 21). The arcuate shapes of the vertical laminations are very pronounced, and the internal strain probably exceeds 100% everywhere, even in the early stage of growth shown in Fig. 21A. The different strain patterns in the two model series can be used to interpret the nature of the viscosity contrast between buoyant material and overburden in natural gneiss domes and salt domes. Natural gneiss domes have been described in detail in the literature, e.g. by Rosenfeld (1968), Thompson et al. (1968) and Naylor (1968). In general, gneiss domes consist of massive acid to intermediate gneisses in the core and thinly laminated and foliated schists and gneisses in the overburden. The overburden rocks typically exhibit large internal strains, as recorded by rotated garnets etc. (Rosenfeld, 1968), whereas the core rocks often appear to be only slightly deformed (Naylor, 1968; and R.A. Cliff quoted by Fletcher, 1972). It therefore seems likely that the first series of models presented here best apply to mantled gneiss domes. The viscosity contrast between core and mantle in the first series of models was only 4.3, and this figure also describes roughly the ratio between the average strain in the overburden and buoyant layer. In natural domes the ratio of the strains is probably somewhat higher, indicating that the viscosity contrast is also higher. Rosenfeld (quoted by Fletcher, 1972) has suggested that it may be as high as 100 in the Chester Dome of Vermont. The sense of rotation within elements in the overburden layer of the first series of models also agrees with that recorded by the garnet rotations in the Chester Dome overburden (Rosenfeld, 1968). This supports the intrastratal flow model proposed by Rosenfeld and also by Fletcher (1972). In contrast with gneiss domes, salt domes exhibit larger strains within the

121

buoyant salt layer than within the sedimentary cover. It is clear that salt has a viscosity several orders of magnitude lower than that of the overlying sediments. Salt domes contain very high-amplitude flow folds indicating that the salt undergoes severe intrastratal flow during the formation of the domes (see the classic description in Balk, 1949). The overlying sediments on the other hand generally undergo virtually no bulk strain. The strata are merely bowed up and overturned as the diapir penetrates upward through them. They commonly exhibit brittle failure in that they are complexly faulted. This is the type of strain pattern found in the second series of models, in which the viscosity contrast between buoyant layer and overburden was 0.23. Thus these models are representative of salt diapirism. Plane flow vs. radial flow The models described in this paper, particularly the first series, were constructed in such a way that they formed cylindrical ridges. This means that the flow during deformation was constrained to two dimensions in the plane normal to the ridge axis. This limitation was necessary in order that the strain could be analyzed (see Dixon, 1974). Natural domes are commonly elliptical or circular in plan and form under conditions of radial flow. Thus the quantitative measures of the strain distribution derived from the models are not strictly representative of the strains within natural radial domes. In estimating the nature of the fabric formed within an element in a natural radial dome, it is necessary to interpret the plane-strain data from the models in the light of this difference. The interpretation can be made qualitatively on the basis of the following reasoning. In the models the plane elements are deformed in two dimensions under conditions of constant area. In contrast, when material moves radially inward toward the center of a dome it is compressed in the direction parallel to the circumference of the dome. Consequently, due to conservation of volume the element increases in area when observed in a radial section. Similarly, material which moves radially outward from the dome becomes stretched parallel to the circumference and the element decreases in area in a radial section. The change in area is in direct proportion to the amount of movement along the radius. The changes in area also have an effect on the shapes of the elements in radial section. Thus it is a non-trivial problem to interpret the strain in radial flow from the strain in plane flow. Some aspects of the modification of the fabrics may be readily seen. For example, the vertical stretching in the core of model WD-4 was interpreted to produce a vertical foliation under conditions of plane flow. If the flow is instead radial, then this vertical stretching would be manifested as a vertical lineation. Again, the elements in the buoyant layer beneath the rim syncline of models WD-8 and WD-3 were interpreted to exhibit a horizontal foliation under conditions of plane flow. If the flow was radially convergent these elements would have instead a horizontal radial lineation. Third, elements in

122

the overburden which move outward from the dome during the deformation would undergo circumferential stretching if the flow was radial, and their fabrics would therefore be modified by some amount of circumferential rodding. This fabric is observed in the Valhalla gneiss dome of British Columbia (Reesor, 1965). Thus the fabrics of elements within a radial dome can be qualitatively estimated from the plane-strain models. CONCLUSIONS

Experimental models of a diapiric ridge can be used to determine the distribution of finite strain within small elements of the structure at various stages in its development. The finite strain states at each stage define points on the deformation path of each element. The path can be used to interpret the nature of the internal fabric of corresponding elements in natural diapiric ridges. The strain distribution within the models supports the contention that in mantled gneiss domes the buoyant core rocks have a higher effective viscosity than the heavy over-burden rocks, and that in salt diapirs the reverse is true. The strains determined from the plane-flow models must be modified before they can be applied to the interpretation of the fabrics in radially symmetric domes. ACKNOWLEDGEMENTS

This paper reports a portion of the author’s doctoral dissertation research, done under the supervision of Professor Hans Ramberg at the University of Connecticut. The experimental model work was carried out at the Tectonics Laboratory of the University of Uppsala, Sweden, during a seven-month visit supported by a National Research Council of Canada Postgraduate Scholarship and a National Science Foundation Doctoral Dissertation research grant. The University of Connecticut Computer Center provided computer facilities and time.

REFERENCES Anketell, J.M., Cegla, J. and Dzulynski, S., 1970. On the deformational structures in systems with reversed density gradients. Ann. Sot. Geol. Poland, 40: 3-30. Balk, R., 1949. Structure of Grand Saline salt dome, Van Zandt County, Texas. Bull. Am. Assoc. Pet. Geol., 33: 1791-1829. Berner, H., Ramberg, H. and Stephansson, O., 1972. Diapirism in theory and experiment. Tectonophysics, 15: 197-218. Biot, M.A. and Ode, H., 1965. Theory of gravity instability with variable overburden and compaction. Geophysics, 30: 213-227. Braunstein, J. and O’Brien, G.D. (Editors), 1968. Diapirism and Diapirs. Am. Assoc. Pet. Geol., Mem., 8, 444 p.

123 Bridgwater, D., Sutton, J. and Watterson, J., 1974. Crustal downfolding associated with igneous activity. Tectonophysics, 21: 57-77. Dane& Z.F., 1964. Mathematical formulation of salt-dome dynamics. Geophysics, 29: 414-424. Dixon, J.M., 1974. A new method for the determination of finite strain in models of geological structures. Tectonophysics, 24: 99-114. Dobrin, M.B., 1941. Some quantitative experiments on a fluid salt-dome model and their geological implications. Trans. Am. Geophys. Union, 22: 528-542. Elliott, D., 1972. Deformation paths in structural geology. Geol. Sot. Am. Bull., 83: 2621-2638. Escher, B.G. and Kuenen, P.H., 1929. Experiments in connection with salt domes. Leidse Geol. Meded., 3: 151-182. Eskola, P.E., 1949. The problem of mantled gneiss domes. Geol. Sot. London, Q. J., 104: 461-476. Fletcher, R.C., 1972. Application of a mathematical model to the emplacement of mantled gneiss domes. Am. J. Sci., 272: 197-216. Fyson, W.K., 1971. Fold attitudes in metamorphic rocks. Am. J. Sci., 270: 373-382. Hubbert, M.K., 1937. Theory of scale models as applied to the study of geological structures. Bull. Geol. Sot. Am., 48: 1459-1520. Naylor, R.S., 1968. Origin and regional relationships of the core-rocks of the Oliverian domes. In: Zen, E-an, et al. (Editors), Studies of Appalachian Geology; Northern and Maritime. Wiley-Interscience, New York, p. 231-240. Nettleton, L.L., 1934. Fluid mechanics of salt domes. Bull. Am. Assoc. Pet. Geol., 18: 1175-1204. Ode, H., 1966. Gravitational instability of a multilayered system of high viscosity. Verh. K. Ned. Akad. Wet., Reeks 1, 24: 96. Parker, T.J. and McDowell, A.N., 1955. Model studies of salt dome tectonics. Bull. Am. Assoc. Pet. Geol., 39: 2384-2470. Price, R.A., 1972. The distinction between displacement and distortion in flow, and the origin of diachronism in tectonic overprinting in erogenic belts. 24th Int. Geol. Congr., Sect. 3, p. 545-551. Ramberg, H., 1963. Experimental studies of gravity tectonics by means of centrifuged models. Bull. Geol. Inst. Univ. Upps., 42: l-97. Ramberg, H., 1967. Gravity, Deformation and the Earth’s Crust, as Studied by Centrifuged Models. Academic Press, London, 214 p. Ramberg, H., 1968a. Fluid dynamics of layered systems in the field of gravity. Phys. Earth Planet. Inter., 1: 63-87. Ramberg, H., 1968b. Instability of layered systems in the field of gravity, I. Phys. Earth Planet. Inter., 1: 427-447. Ramberg, H., 1971. Dynamic models simulating rift valleys and continental drift. Lithos, 4: 259-276, Ramberg, H., 1972. Mantle diapirism and its tectonic and magmagnetic consequences. Phys. Earth Planet. Inter., 5: 45-60. Ramberg, H. and Sjijstrom, H., 1973. Experimental geodynamical models relating to continental drift and orogenesis. Tectonophysics, 19: 105-132. Ramsay, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, London, 568 p. Ramsay, J.G. and Graham, R.H., 1970. Strain variation in shear belts. Can. J. Earth Sci., 7: 786-813. Reesor, J.E., 1965. Structural evolution and plutonism in Valhalla gneiss complex, British Columbia. Geol. Surv. Canada, Bull. 129. Roberts, D., 1971. Abnormal cleavage patterns in fold hinge zones from Varanger Peninsula, northern Norway. Am. J. Sci., 271: 170-180. Roberts, D. and StrSmgard, K.-E., 1972. A comparison of natural and experimental strain patterns around fold hinge zones. Tectonophysics, 14: 105-120.

124 Rosenfeld, J.L., 1968. Garnet rotations due to the major Paleozoic deformations in southeast Vermont. In: Zen, E-an, et al. (Editors), Studies of Appalachian Geology: Northern and Maritime. Wiley-Interscience, New York, p. 185-202. Schwerdtner, W.M., 1973. Schistosity and penetrative mineral lineation as indicators of paleostrain directions. Can. J. Earth Sei., 10: 1233-1243. Sdig, F., 1965. A theoretical prediction of salt dome patterns. Geophysics, 30: 633-643. Stephansson, O., 1971. Gravity tectonics on Gland. Bull. Geol. Inst. Univ. Upps., New Ser., 3: 31-78. Stephansson, O., 1972. Theoretical and experimental studies of diapiric structures on ijland. Bull. Geol. Inst. Univ. Upps., New Ser., 3: 163-200. Taylor, G.I., 1950. Instability of liquid surfaces when accelerated in a direction perpendicular to their planes, I. Proc. R. Sot. (London), Ser. A, 210: 192-196. Thompson, J.B., Jr., Robinson, P., Clifford, T.N. and Trask, N., 1963. Nappes and gneiss domes in west-central New England. In: Zen, E-an, et al. (Editors), Studies of Appalachian Geology: Northern and Maritime. Wiley-Interscience, New York, p. 203-218. Van Bemmelen, R.W., 1972. Geodynamic Models. Elsevier, Amsterdam, 267 p.