6. Deformation and Strain - Supplements
Contents
Mohr Circle Construction for Strain
SUPPORTING MATERIAL
Types of Strain
Coaxial strain Strain in which the incremental strain axes remain parallel to the finite strain axes during progressive strain.
Heterogeneous strain Strain in which any two portions of a body similar in form and orientation before strain undergo relative change in form and orientation (also: inhomogeneous strain).
Homogeneous strain Strain in which any two portions of a body similar in form and orientation before strain remain similar in form and orientation after strain.
Incremental strain Strain state of one step in a progressive strain history.
Instantaneous strain Incremental strain of vanishingly small magnitude (a mathematical descriptor); also called infinitesimal incremental strain.
Finite strain Strain that compares the initial and final strain configurations; sometimes called total strain.
Non-coaxial strain Strain in which the incremental strain axes rotate relative to the finite strain axes during progressive strain.
MORE EXPLORATION
Basic Strain Calculation
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Consider the geologic situation that’s is sketched on the right (Figure S6.1). A sequence of tilted sandstone beds (grays) is unconformably overlain by a unit containing ellipsoidal inclusions (say, clasts in a conglomerate). The strain ratio of the inclusions in sectional view is X/Y = 4, and the dip of the underlying beds is 50o. What was the angle of dip of the sandstone beds in sectional view if the inclusions were originally spherical? |
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Mohr Circle Construction for Strain
We learned that the strain state is described geometrically by an ellipsoid, so strain is a second-rank tensor. We can use the same mathematics for strain that we earlier used for stress, but do remember that the stress and strain ellipsoids are not the same. Because of the mathematical similarities, a Mohr construction for strain can be used to represent the relationship between longitudinal and angular strain in a manner similar to that for σn and σs in the Mohr diagram for stress. Usually the quadratic elongation, λ, and the shear strain, γ, are used, for which we need to rewrite some relationships and introduce a few convenient substitutions.
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FIGURE S6.2 The Mohr construction for strain. For a deformed object (a), the reciprocal values of the principal strains are plotted in γ'-λ' space, where λ' = 1/λ and γ' = γ/λ (b). The corresponding rotation of line OP in XZ-space (or λ1-λ3 space) is shown in (c). [4.13] |
Considering Figure S6.2c and applying some trigonometric relationships, we get:
λ = λ1 cos2 ϕ + λ3 sin2 ϕ Eq. S6.18
λ = ½ (λ1 + λ3 ) + ½ (λ1 - λ3 ) cos 2ϕ Eq. S6.19
and
γ = (λ1/λ3 – λ3/λ1 – 2)1/2 cos ϕ sin ϕ Eq. S6.20
γ = -½ (λ1 – λ3) sin 2ϕ Eq. S6.21
But this expresses strain in terms of the undeformed state. We observe a body after strain has occurred, so it is more logical to express strain in terms of the deformed state. We therefore need to express the equations in terms of the angle ϕ' that we measure rather than the original angle ϕ, which is generally unknown. To this end we introduce the parameters λ' = 1/λ and γ' = γ/λ and use the equations for double angles. We then get:
λ' = ½ (λ1’ + λ3’ ) - ½ (λ3’ – λ1’ ) cos 2ϕ’ Eq. S6.22
γ’ = ½ (λ3’ – λ1’) sin 2ϕ’ Eq. S6.23
If you compare these equations with those for the normal stress (σn) and the shear stress (σs) and follow their manipulation, you will find that Equations S6.22 and S6.23 describe a circle with a radius (λ3' – λ1'), whose center is located at (λ1' + λ3') in a reference frame with γ' on the vertical axis and λ' on the horizontal axis. This is the Mohr circle construction for strain.
At first glance these manipulations appear unnecessarily confusing and they tend to discourage the application of the construction. So, let’s look at an example (Figure S6.2). Assume that a unit square is shortened by 50% and extended by 100% (Figure S6.2a). Thus, e1 = 1 and e3 = –0.5, respectively; consequently, λ1 = 4 and λ3 = 0.25. Note that the area remains constant because λ11/2 ⋅ λ31/2 = 1. Using the parameter λ', we get λ1' = 0.25 and λ3' = 4. Plotting these values on the Mohr diagram results in a circle with radius r = (λ3' – λ1') = 1.9, whose center is at (λ1' + λ3') = 2.1 on the λ' axis. It is now quite simple to obtain a measure of the longitudinal strain and the angular strain for any line oriented at an angle ϕ' to the strain axes. For example, for a line in the λ1λ3-plane (i.e., XZ-plane) of the strain ellipsoid at an angle of 25o to the maximum strain axis, we plot the angle 2ϕ' (50o) from the λ1' end of the circle and draw line OP' (Figure S6.2b). The corresponding strain values are:
λ' = 0.9 and γ' = 1.4, thus λ = 1.1 and γ = 1.5
This means that if line OP represented the long axis of a fossil (e.g., a belemnite), it will have extended and also rotated from this original configuration. Using Equation 4.10 (applied in Figure S6.2), we can also calculate the original angle ϕ that our belemnite made with the reference frame:
ϕ = arctan [(λ1/λ3)1/2 ⋅ tan ϕ'] = 62o
This latter calculation highlights the easily misunderstood relationship between the angular shear and the rotation angle of a particular element in a deforming body. The rotation of line OP to OP' in the deformed state occurred over an angle of 37o (62o – 25o). However, this angle is not equal to the angular shear, ψ, of that element, gives ψ = 56o. We plot these various angles in λ1λ3- space (i.e., XZ-space) in Figure S6.2c.
You may have noticed that we consider coaxial strain in our example of the Mohr circle for strain construction, in which the incremental strain axes are parallel to the finite strain axes. The construction for non-coaxial strain adds a component of rotation to the deformation. Note that this rotation is different from the line rotation in our example; all lines, except material lines that parallel the principal strain axes, rotate in coaxial strain. In Mohr space, this rotational component moves the center of the Mohr circle off the λ' (reciprocal longitudinal strain) axis. In fact, the rotational component of strain can be quantified from the off-axis position of the Mohr circle, but this application takes us well beyond an introduction to the Mohr construction for strain.
Logarithmic Strain Plot
A useful modification of the Flinn diagram for strain, called the Ramsay diagram[1], uses the natural logarithm of the values a and b (Figure S6.3):
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FIGURE S6.3. The Ramsay diagram plots the logarithm of strain ratios X/Y and Y/Z; in the case of the natural logarithm (ln) it plots (ε1 – ε2) versus (ε2 – ε3). The parameter K describes the shape of the strain ellipsoid. Note that volume change, denoted by Δ, produces a parallel shift of the line K = 1. Thus, we recognize fields of apparent constriction and flattening. [4.16] |
ln a = ln (X/Y) = ln [(1 + e1)/(1 + e2)] Eq. S6.27
ln b = ln (Y/Z) = ln [(1 + e2)/(1 + e3)] Eq. S6.28
Using
ln x/y = ln x – ln y and ε = ln (1 + e)
where ε is natural strain, we convert Equations 4.27 and 4.28 to:
ln a = ε1 – ε2 Eq. S6.29
ln b = ε2 – ε3 Eq. S6.30
The parameter k of the Flinn plot becomes K in the Ramsay diagram:
K = ln a/ln b = (ε1 – ε2)/(ε2 – ε3) Eq. S6.31
Volume change expressed in terms of natural strains, becomes:
ln (Δ + 1) = ε1 + ε2 + ε3 Eq. S6.33
Further rearrangement of this expression in a form that uses the axes of the Ramsay diagram gives:
(ε1 – ε2) = (ε2 – ε3) – 3ε2 + ln (Δ + 1) Eq. S6.34
Prolate and oblate ellipsoids are separated by plane strain conditions, where ε2 = 0:
(ε1 – ε2) = (ε2 – ε3) + ln (Δ + 1) Eq. S6.35
Both logarithmic plots with base e (natural logarithm, ln) and base 10 (log) are used in the Ramsay diagram. The Ramsay diagram is similar to the Flinn diagram in that the line K = 1 separates the fields of constriction (∞ > K > 1) and flattening (1 > K > 0) and the unit sphere lies at the origin (ln a = ln b = 0). Note that the origin in the Ramsay diagram has coordinates (0, 0). There are a few advantages to the Ramsay diagram. First, small strains that plot near the origin and large strains that plot away from the origin are more evenly distributed. Second, the Ramsay diagram allows a graphical evaluation of the incremental strain history, because equal increments of progressive strain (the strain path) plot along straight lines, whereas unequal increments follow curved trajectories. In the Flinn diagram both equal and unequal strain increments plot along curved trajectories.
READING
Elliott, D., 1972. Deformation paths in structural geology. Geological Society of America Bulletin, 83, 2621–2638.
Erslev, E. A., 1988. Normalized center-to-center strain analysis of packed aggregates. Journal of Structural Geology, 10, 201–209.
Fry, N., 1979. Random point distributions and strain measurement in rocks. Tectonophysics, 60, 89–104.
Groshong, R. H., Jr., 1972. Strain calculated from twining in calcite. Geological Society of America Bulletin, 83, 2025–2038.
Lisle, R. J., 1984. Geological strain analysis. A manual for the Rf /Φ method. Pergamon Press: Oxford.
Lister, G. S., and Williams, P. F., 1983. The partitioning of deformation in flowing rock masses. Tectonophysics, 92, 1–33.
Means, W. D., 1976. Stress and strain. Basic concepts of continuum mechanics for geologists. Springer- Verlag: New York.
Means, W. D., 1990. Kinematics, stress, deformation and material behavior. Journal of Structural Geology, 12, 953–971.
Oertel, G., 1983. The relationship of strain and preferred orientation of phyllosilicate grains in rocks— a review. Tectonophysics, 100, 413–447.
Pfiffner, O. A., and Ramsay, J. G., 1982. Constraints on geologic strain rates: Arguments from finite strain states of naturally deformed rocks. Journal of Geophysical Research, 87, 311–321.
Ramsay, J. G., and Wood, D. S., 1973. The geometric effects of volume change during deformation processes. Tectonophysics, 16, 263–277.
Ramsay, J. G., and Huber, M. I., 1983. The techniques of modern structural geology. Volume 1: Strain analysis. Academic Press: London.
Simpson, C., 1988. Analysis of two-dimensional finite strain. In Marshak, S., and Mitra G., eds., Basic methods of structural geology. Edited by: S. Marshak and G. Mitra. Prentice Hall: Englewood Cliffs, pp. 333–359.