7. The Plastic Regime - Supplements
Terminology of Crystal Plasticity and Diffusion
Dominant Slip Systems In Common Rock-Forming Minerals
Summary Table of Deformation Mechanisms and Processes
Constructing a Deformation Mechanism Map
SUPPORTING MATERIAL
Terminology of Crystal Plasticity and Diffusion
Annealing Loosely used term for high-temperature grain adjustments, including static recrystallization and grain growth.
Bulge nucleation A type of migration recrystallization in which a grain boundary bulges into a grain with higher internal strain energy, forming a recrystallized grain.
Dislocation wall Concentration of dislocations in a planar array.
Dynamic recrystallization Formation of relatively low-strain grains under an applied differential stress.
Foam structure Recrystallized grain structure characterized by the presence of energetically favorable grain-boundary triple junction (at ≈120° angles).
High-angle boundary Boundary across which the crystallographic mismatch exceeds 10°; characteristic of recrystallization.
Low-angle boundary Tilt boundary across which the crystallographic mismatch is less than 10°; characteristic of recovery.
Migration recrystallization Recrystallization mechanism by which grain boundaries move driven by a contrast in strain energy between neighboring grains.
Polygonized microstructure Recovery structure showing elongate to blocky subgrains (mostly used for phyllosilicates).
Recovery Process that forms low-angle grain boundaries by the temperature- activated rearrangement of dislocations.
Recrystallization Mechanism that removes internal strain energy of grains remaining after recovery, producing high-angle grain boundaries that separate relatively strain-free (recrystallized) grains.
Recrystallized grains Relatively low-strain grains that are formed by recrystallization.
Rotation recrystallization Recrystallization mechanism by which dislocations pile up in a tilt boundary, thereby “rotating” the crystal lattice of the area that is enclosed by the tilt boundary.
Static recrystallization Formation of strain-free grains after deformation has stopped (i.e., differential stress is removed).
Subgrain Area of crystallographic mismatch that is less than 10° relative to the host grain.
Subgrain rotation Recrystallization mechanism by which dislocations continue to move into a low- angle tilt boundary surrounding a subgrain, thereby increasing the crystallographic mismatch and forming a high-angle grain boundary.
Superplastic creep Grain-size-sensitive deformation mechanism by which grains are able to slide past one another without friction because of the activity of diffusion (as opposed to frictional sliding or cataclasis).
Tilt boundary Concentration of dislocations in a planar array.
Twinning Deformation mechanism that rotates the crystal lattice over a discreet angle such that the twin boundary becomes a crystallographic mirror plane. Such a planar defect is produced by the motion of partial dislocations.
Undulose extinction Irregular distribution of dislocations in a grain, producing small crystallographic mismatches or lattice bending that is visible under crossed polarizers.
Dominant Slip Systems In Common Rock-Forming Minerals
Mineral Glide plane + Slip directiona Comments
Calcite {-1018} <40-41> e-twinning
{10-14} <-2021> r-twinning
{10-14} <-2021> r-glide
{01-12} <2-201> or <-2021> f-glide
Dolomite {-1012} <10-11> f-twinning
(0001) <2-1-10> c-glide
{01-12} <2-201> or <-2021> f-glide
Mica (001) <110> basal (c) slip
Olivine (001) [100] {110} [001]
Quartz (0001) <11-20> basal (c) slip
{10-10} [0001] prism (m) slip, along c
{10-10} <11-20> prism (m) slip, along a
{10-11} <11-20> rhomb (z) slip
aMiller indices for equivalent glide planes from crystal symmetry are indicated by { }; specific glide planes are indicated by ( ); equivalent slip directions from crystal symmetry are indicated by < >; individual slip directions are indicated by [ ]. From: Wenk, 1985
Summary Table of Deformation Mechanisms and Processes
From Jessell and Bons, 2002.
MORE EXPLORATION
Calcite Twinning Analysis
The fact that twinning takes place along specific crystallographic planes in a calcite crystal,1 and that rotation occurs over a specific angle and in a specific sense, allows us to use twinning as a measure of finite strain and differential stress.
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FIGURE S7.1. Calcite strain gauge technique. An original grain ABCD (a) with a single twin of thickness, t (shaded region). In (b) a grain with multiple twins (shaded regions) is shown. [9.19] |
In Figure S7.1a, a deformed grain A′B′CD with one twin is shown; the original grain outline is ABCD, whose sides are parallel to calcite crystal planes. From this figure you can see that the shear strain for the twinned grain is:
γ = tan ψ = q/T Eq. S7.2
For one twin,
q = p, so γ = [2t tan(φ/2)]/T Eq. S7.3
where T is the grain thickness and t is the twin thickness. For a grain containing several twins (Figure 9.18b) thes hear strain is obtained by adding the strain due to each twin, or:
Eq. S7.4
where n is the number of twins in the grain. Given that the angle φ is constant in the case of calcite (≈38°; Figure 9.18b), Equation 9.4 simplifies to:
Eq. S7.5
So, if we measure the total width of twins and the grain size perpendicular to the twin plane we can obtain the total shear strain for a single twinned grain. In an aggregate of grains, the shear strains will vary as a function of the crystallographic orientation of individual grains relative to the bulk strain ellipsoid, and we use this variation to determine the orientation of the principal strain axes by determining the orientations for which the shear strains are zero and maximum. This strain analysis technique is called the calcite strain-gauge method. Looking again at Figure S7.1 and Equation S7.5, we can now determine the maximum amount of shear strain that can be accumulated using twinning: γmax occurs when the entire grain is twinned, so t = T; thus, γmax = 0.7, or X/Z ≈ 2. This maximum contrasts with the amount of strain that can accumulate during dislocation glide, which is unrestricted. Also, methods for the determination of the differential stress for an aggregate with twinned grains have been developed that use the number of activated twin planes. Thus, calcite twinning analysis can give both strain and differential stress magnitudes from naturally deformed carbonates. The method has proved to be very useful in studying stress and strain fields in limestones that were subjected to small strains, the kinematics of folding, the formation of veins, the early deformation history of fold-and-thrust belts, and deformation patterns in continental interiors.
Constructing a Deformation Mechanism Map
The value of deformation mechanism maps is best understood and appreciated when you construct your own. Below we list constitutive equations for various deformation mechanisms in natural limestones and marbles, which will allow you to construct a deformation mechanism map (Figure S7.2).
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FIGURE S7.2. Experimentally derived constitutive equations used for the construction of a deformation mechanism map for calcite at T = 475°C. The thin lines represent strain rates (marked as –log), whereas the thick lines separate deformation mechanism fields. [9.37] |
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First, choose the axes of the plot. We’ll plot differential stress versus temperature. Now calculate the corresponding strain rate from each of the four constitutive equations at a specific stress and temperature condition (i.e., a point in the diagram). You recall that the mechanism producing the highest strain rate is dominant, so from the four solutions the one with the highest strain rate is dominant at that particular point in the diagram. Using individual points to fill the diagram is an unnecessarily slow and cumbersome approach. Instead, we calculate the stress–temperature curves at a given strain rate for each equation and plot these four curves in the diagram. Because some of these curves intersect, the final strain rate curve is composed of segments of the four curves for which the differential stress is smallest.
When using different strain rates you will see that the positions of intersection points change. Also, the dominant deformation mechanism may change. Connecting these intersection point where mechanisms change defines the boundary between fields. A worked-out example is shown, in which differential stress is plotted as a function of grain size for T = 475°C. You can vary environmental conditions, such as stress, temperature, and grain size, and calculate the corresponding map using fairly simple spreadsheet calculations on a personal computer.
READING
Ashby, M. F., and Verrall, R. A., 1978. Micro mechanisms of flow and fracture, and their relevance to the rheology of the upper mantle. Philosophical Transactions of the Royal Society of London, Series A, 288, 59–95.
Frost, H. J., and Ashby, M. F., 1982. Deformation mechanism maps. The plasticity and creep of metals and ceramics. Pergamon Press: Oxford.
Groshong, R. J., Jr., 1972. Strain calculated from twinning in calcite. Geological Society of America Bulletin, 83, 2025–2038.
Hayden, H. W., Moffatt, W. G., and Wulff, J., 1965. The structure and properties of materials: Volume III, Mechanical behavior. J. Wiley and Sons: New York.
Hull, D., and Bacon, D. J., 1984. Introduction to dislocations. Pergamon Press: Oxford.
Jamison, W. R., and Spang, J. H., 1976. Use of calcite twin lamellae to infer differential stress. Geological Society of America Bulletin, 87, 868–872.
Jessell, M., and Bons, P., 2002. On-line short course in microstructures. http://www.earth.monash.edu.au/ Teaching/mscourse/index.html.
Loretto, M. H., and Smallman, R. E., 1975. Defect analysis in electron microscopy. Chapman and Hall: London.
Mercier, J.-C., Anderson, D. A., and Carter, N. L., 1977. Stress in the lithosphere: inferences from steady-state flow of rocks. Pure and Applied Geophysics, 115, 199–226.
Nicolas, A., and Poirier, J.-P., 1976. Crystalline plasticity and solid state flow in metamorphic rocks. John Wiley and Sons: London.
Ord, A., and Christie, J. M., 1984. Flow stresses from microstructures in mylonitic quartzites of the Moine thrust zone, Assynt area, Scotland. Journal of Structural Geology, 6, 639–654.