1. Folds and Folding 2 Lecture 22 – Spring 2016
Structural Geology
Folds and Folding 2
Lecture 22 – Spring 2016

2. Layering Shows Folding
Deformational forces are necessary to produce folding, but layering must be present in order to see the folding
The properties of the layers make a difference
Two cases, active folding and passive folding, are recognized. In passive folding, the layering has no mechanical significance. Different colored layers in an essentially homogeneous rock will produce passive folding. Elevated temperatures often lead to passive folding, and it is commonly seen in metamorphic rocks.
Figure in text

3. High Homologous Temperature
If the homologous temperature is high (near the melting point) we find migmatites forming
These rocks often have complex folded structures
Source:
Photo: Martin Miller
Death Valley N.P.

4. Glacier Deformation
Similar patterns are seen in glaciers deformed near their melting point
The tongue of the Malaspina Glacier, the largest glacier in Alaska
Image provided by the USGS EROS Data Center Satellite Systems Branch
Click image for large scale photo (3+ Megs!)
The folds seen in passive folding are the amplification of natural imperfections in the rock or ice, or are due to differential flow within the body of the deforming substance.
Active folding is also known as flexural folding. The layers have mechanical significance, which means some layers are stronger, or more competent, than others. The difference in competency can strongly affect the folding pattern. Two cases of active folding are recognized, bending and buckling.
Source:

5. Bending
Figure 10.28a in text
Bending involves the application of force at an oblique angle to the layering
This can occur during the infilling of a basin, when ice accumulates on part of a lithospheric plate, or during the formation of a monoclinal fold

6. Buckling
Figure 10.28b in text
An applied force parallel to the mechanical anisotropy in the rock produces buckling
Collision between lithospheric plates of equal density can produce buckling

7. Competency Effects
Figure illustrates the process of buckling with layers of different competency
A band of rubber is surrounded by foam in a transparent box
A plunger pushes down from the top, acting like a piston. A thicker rubber band is more competent than a thin band
The observed pattern of folding varies significantly depending on the competency of the rubber band
Figure 10.29a,b in text

8. Arclength vs. Thickness
In performing this experiment, we see that the wavelength and arclength increase as the competency of the rubber band grows larger
This relation was observed early in the twentieth century in sandstone
We can also demonstrate that equally thick bands with different degrees of stiffness will affect the folding. The stiffer the band, the larger will be the wavelength and arclength. The folds are permanent strain features, and can be interpreted in terms of different rheologies, both linear and non-linear. For simplicity, we will use Newtonian viscosity, but must recognize that other rheologies are possible
Figure in text

9. Biot - Ramberg Equation
The Biot - Ramberg equation, first work out in the 1960's, relates the arclength(L)-thickness(t) relationship for a layer of viscosity ηL surrounded by matrix with viscosity ηM:
L = 2πt(ηL/6ηM)a
Arclength is directly proportional to the thickness, and to the cube root of the viscosity ratio

10. Viscosity Ratio
We can measure arclength, and thus obtain the viscosity ratio:
ηL/ηM = (L/t)3
Viscosity is proportional to the cube of the arclength to thickness ratio
The data from figure can be used to estimate the viscosity

11. Error Propagation
Because this is a log-log plot, small errors in the (L/t) parameter lead to large errors in the viscosity ratio, and thus we can only estimate to an order of magnitude
Suppose we redo the foam experiment with the rubber, but merely use a marker pen to draw a line on the foam. Compression shortens the line, but does not cause folding. The line is observed to thicken. The ηL/ηM ratio is 1, so we observe that low viscosity contrasts produce strain-induced layer thickening. For our analysis, we simplify by saying that layer thickening occurs before folding. We can modify the equation to include a strain component:
Figure in text

12. Modified Biot – Ramberg Equation
We can modify the equation to include a strain component:
L = 2πt[(ηL(R-1)/6ηM • 2R2)]a
Where R is the strain ratio X/Z
This equation is known as the modified Biot -Ramberg equation, and it works well in rocks with low viscosity contrasts
Many metamorphic rocks fit this category

13. Increasing Viscosity Contrast
Numerical modeling can be used to study the effects of layer thickening and increasing viscosity contrast
Figure shows one such study
Figure in text

14. Diminishing Viscosity Contrast
We can quickly see that as viscosity ratio decreases, so does arclength (look down a column)
We also see that the strain pattern immediately surrounding the layer is more homogeneous at low viscosity contrasts
At higher conditions of temperature and pressure, folding is likely to involve nonlinear rheologies, in which the viscosity is stress dependent.
Studies have shown that nonlinear rheologies result in lower values of L/t, and a negligible component of layer-parallel shortening.
Natural folds in metamorphic terranes have low L/t ratios (L/t < 10). This indicates a power-law rheology is dominant, and that crystal-plastic processes are important under natural conditions.

15. Multilayers
In nature, we are not limited to a single layer, so the discussion needs to be extended to multilayer situations
Adding two layers of identical thickness results in a longer arclength
The two layers seem to act as a single, thicker layer
Figure in text

16. Thick and Thin Layer
Combining a thick and thin layer results in the thick layer behaving similarly to a single thick layer, but the thin layer changing behavior radically
The thin layer mimics the behavior of the thicker layer
Experiments with multi-layer systems show that the thicker layers dominant the behavior. Thicker layers are more competent, and “stiffer”. Thus, the Biot-Ramberg theory applies only when layers are sufficiently far apart.

17. Contact Strain Zone
The region over which the effect of buckling is significant is the contact strain zone (CSZ)
The width of the CSZ is 2 • dCSZ, where dCSZ is measured from the midpoint to the outside of the buckled layer
This is a function of arclength:
dCSZ = 2/π L
In practical terms, the width of the CSZ is slightly greater than the arclength of the fold

18. Overlapping CSZ’s
If layers are within each others CSZ’s, they interact
If we assume that layers are free to move relative to each other, we can extend our earlier mathematics:
Lm = 2πt(NηL/6ηM)a
where N is the number of superposed layers, with infinitely small spacing between them

19. Single Layer vs. Multilayer Arclength
We can show that this is not equivalent to a single layer of thickness Nt
Dividing the single layer equation with thickness Nt by the multilayer equation gives:
LS/Lm = Nb
Thus the arclength of N layers of thickness t is less than that of a single layer whose thickness is Nt
The multilayer equation is not applicable to many geologic systems. It supposes that the competencies of each layer are the same.
Commonly, we find layers with different viscosities alternating. One example would be a turbidite layering of sandstone and shale. The analysis, especially the required assumptions, become increasing complex, but the general observation is that multilayer systems behave like thick single layers, but the arclength in the multilayer system is less than in the single layer system.

20. Kinematic Models of Folding
The models are:
Flexural Folding
Neutral-surface folding
Shear folding
Fold Shape Modification
Four models of the actual mechanisms of folding have been developed, each having distinct properties.
Each produces folds with different characteristics.
This allows us to compare the models with natural examples to see if they offer an explanation for observed behavior

21. Flexural Folding Diagram
Figure in text
Folds that form by slippage between layers are called flexural slip folds
The amount of slip increases away from the hinge zone, and reaches a maximum at the inflection point
The amount of slip is proportional to the dip angle of the limb
Figure illustrates the concept

22. Characteristics of Flexural Folding
Three important characteristics associated with flexural folding are:
At each point in the profile plane the strain ratio and orientation differ
The 3-D strain state of the fold is plane strain X > Y = 1 > Z, and Y is parallel to the hinge line
The fold is cylindrical, and belongs to class 1B (parallel) - bed thickness in flexural folds does not change
The geometrical consequences of this type of folding are shared by neutral-surface folds. Kink folds, and Chevron folds are examples of flexural slip folds, so are not diagnostic
A related type of folding is flexural flow folding. In this case, slip occurs on individual grains within a layer, leaving no visible slip surfaces. The geometrical and kinematic characteristics of both flexural slip and flexural flow folding are identical.

23. Neutral-Surface Folding Diagram
Figure shows the strain pattern for neutral-surface folding
The top of the folded layer is stretched, and the long axis of the strain ellipse is perpendicular to the hinge line
Neutral-Surface Folding
When a solid object like a metal bar is bent, there are no layers to slip past each other. The metal itself is deformed. Some rocks behave the same way. The fold shape produced is parallel and cylindrical, just as it was with flexural folding. The intermediate strain axis is parallel to the fold axis, like flexural folding. The strain patterns for the two cases differ, however.
Figure in text

24. Compression and Elongation
The bottom layer is compressed, so the long axis of the strain ellipse is now parallel to the hinge line
Clearly, somewhere in between the top and the bottom, there is a layer which is neither stretched not compressed
At this layer, the strain ellipse remains a perfect sphere. In the fold profile plane, the unaffected layer is called the neutral-surface, which gives this type of folding its name
Strain also accumulates in other surfaces. On the top surface of the upper layer, a linear feature which makes an initial angle α with the fold axis will be stretched, and the angle will increase. On the lower surface, the line will be compressed, and the angle will decrease. Thus, the angle between flute marks and the hinge line increases on the upper surface, and decreases on the lower surface.
The neutral surface does not have to be midway between the upper and lower surfaces. It is usually somewhere between the top and bottom surfaces, but may reach the outer arc in a few cases.

25. Shear Folding
Shear folding is a type of passive folding, where the layers have no mechanical significance
Material flows as a continuum, such as in a glacier
Rock, under certain circumstances, can also flow this way
Source: Source:

26. Shear Folding Example
Imagine a deck of cards with a line drawn obliquely across the side of the deck
If the cards are moved randomly, we produce a fold
The length of the line on each card remains the same, and the dip isogons are parallel
Figure in text

27. Shear Folding Diagram
The fold shape will vary, depending on how each card is displaced relative to its neighbors
Overall strain is plane strain
The hinge line is not parallel to the intermediate strain axis, as it was with the first two types of folding
Fold shape modification
As parallel folds develop, they become tighter and tighter. There are physical limits on the degree to which folds can tighten. If the inner layers can accommodate the compression, folding can continue. But if all layers are equally competent, late stage strain increments will begin to affect the entire fold structure, including the hinge and the limbs. The result is superimposed homogeneous strain.

28. Fold Shape Modification Diagram
Figure shows the modification of the fold shape
Flexural folds and neutral-surface folds are shown with a superimposed homogeneous strain of 20% (that is, with X/Z = 1.6) and 60% (X/Z = 6.3)
The limbs thin relative to the hinge, and the initially parallel fold approaches a condition of being a similar fold. In order to achieve similar fold status, X/Z must be infinite, which is not achievable, but the class 1B folds are now class 1C folds.
We still have no mechanism for producing class 3, or divergent, folds. Yet class 3 folds exist in nature. One case we have not considered is that with layers of strongly contrasting competencies.
Figure in text

29. Sandstone and Shale Diagram of the classic sandstone-shale combination
Figure in text
Diagram of the classic sandstone-shale combination
The shale is much less competent than the sandstone, and accommodates the strain when the layers are folded
Sandstone layers are stippled

30. Effect of Layer Spacing
When sandstone layers are closely spaced, class 3 folds develop
When they are far apart, class 2 folds are formed

31. H.C. Sorby
In the nineteenth century, Henry Clifton Sorby recognized the effect of differing competencies on fold geometries
The sketch shows a folded sandstone layer in slate
Strain in the slate is homogeneous except near the sandstone
Folding in sandstone is class 1C
Sorby is often credited with inventing the petrographic microscope – a biography of Sorby is available at

32. Pebbles in Limestone
Figure in text
Figure shows a diagram of this pebbles in limestone
The observed pattern (10.37a) looks more like neutral-surface folding(10.37d) than flexural folding (10.37c)
However, the magnitude of the strain ratios is too low in the neutral-surface pattern

33. Fold Modification Fold modification might provide an answer (10.37b)
Prior to folding, compaction with shortening perpendicular to the layers occurs, and is followed by neutral-surface folding, during which material is preferentially removed from the inner arc, which gives pattern very similar to the natural pattern
Dissolution and material transport during folding are common in rocks. Quartz and calcite veins in folded rocks give evidence of the transport.

34. Bulk Shortening Strain
Bulk shortening strain can be estimated using the equation:
e = (Lw - La)/La
where Lw is the wavelength and La is the arclength
If the clasts are more competent than the matrix, which is highly likely, the equation is an underestimate of actual strain

35. Folding Scenario
Initial layer a is compacted by 20% (b), followed by layer-parallel shortening (c), and buckling (d)
Homogenous shortening strain (e) transforms a parallel fold into a similar fold
Figure presents a possible sequence of events for a single layer fold. After deposition of a bed, compaction drives out fluid, compacting the bed perpendicular to the layering by 20%. (Xc/Zc = 1.25) Natural systems usually show compaction ranges of 0 to 50%.
The system is then compressed, with layer parallel shortening (LPS) amounting to 20%. (10.38c), This represents a constant volume strain component, which restores the strain ellipse to a circle. (Xf/Zf = 1.00) The corresponding LPS strain ratio is 1.25 (10.38c).
Continued shortening results in initiation and growth of a parallel fold by flexural folding. The arc length, L, is a function of the thickness, t, and the linear viscosity ratio, ηL/ηM. The viscosity of the system may be estimated by measuring L/t and the strain ration Xlps/Zlps.
As buckling begins, strain changes from homogeneous to heterogeneous. Coaxial strain dominates the hinge area, and non-coaxial strain dominates the limb area. (10.38d)
Continued shortening is through superimposed homogeneous strain (10.38e). The end result is a similar fold. The strain pattern varies as a function of compactional strain, the operative field mechanism, the viscosity ratio, and the degree of superimposed homogeneous strain.
Figure in text