deformation and strain - impacttectonics.org · 2014. 11. 30. · finite strain (difference between...

Deformation and Strain

Earth Structure (2nd Edition), 2004

W.W. Norton & Co, New York

Slide show by Ben van der Pluijm

© WW Norton, unless noted otherwise

Lecture 4

© EarthStructure (2nd ed) 29/23/2014

Strain


Deformation is:

b) Strain (distortion)

• Extension (or stretch)length changes

• Internal rotation (vorticity)finite strain axes rotate relative

to instantaneous strain axes

• Volume change

c) Rigid-body rotation (or spin)instantaneous axes and finite axes rotate

together

d) Rigid-body translation

© EarthStructure (2nd ed)4

9/23/2014

Homogeneous vs. heterogeneous strain

Heterogeneous strain (c) is produced by variable

slip on the cards, for example by increasing the slip

on individual cards from bottom to top.

(deck of cards) FIGURE 4.3 A square and a circle

drawn on a stack of cards

Homogeneous strain (b) transform into a

parallelogram and an ellipse

�Straight lines remain straight�Parallel lines remain parallel�Circles become ellipses (or spheres become ellipsoids)


Principal strain axes

Homogeneous strain: transformation of square to rectangle, circle to ellipse.

Two material lines perpendicular before and after strain are the principal axesof the strain ellipse (red lines). The dashed lines are other material lines that

do not remain perpendicular after strain; they rotate toward long axis of strain

ellipse.


Strain path

� Incremental strain (strain steps)

� Finite strain (difference between unstrained and final shape)

� (Infinitesimal strain)

FIGURE 4.5 The finite strains, Xf and Yf, in (a) and (b) are the same, but the strain path by which each was reached is different. This illustrates the importance of understanding the incremental strain history (here, Xi and Yi) of rocks and regions and inherent limitation of finite

strain analysis.


Progressive Strain - Basics

DePaor, 2002



DePaor, 2002


Non coaxial and coaxial strain

• The incremental strain axes are different material lines during non-coaxial strain

• In coaxial strain the incremental strain axes are parallel to the same material lines.

• Note that the magnitude of the strain axes changes with each step.

(progressive) simple shear or non-coaxial strain

(progressive) pure shear, or coaxial strain


Vorticity (internal rotation) and Particle paths

Kinematic vorticity number:

Wk = cos α

a. Wk = 0 (pure shear)

b. 0 < Wk < 1 (general shear)

c. Wk = 1 (simple shear)

d. Wk = ∞ (rigid-body rotation)

FIGURE 4.8 Particle paths or flow lines during

progressive strain accumulation.

The cosine of the angle alpha is the kinematic

vorticity number, Wk, for these strain histories; Wk = 0, 0 <Wk < 1,Wk = 1, and Wk=∞,

respectively.

general shear simple shear pure shear rigid-body rotation


Vorticity (internal rotation) and Particle paths


Superimposed strain

Practically this means that structures reflecting different strain states may be formed, especially in non-coaxial strain regimes such as shear zones.

(a) Coaxial superimposition of a strain event 2 produces regions in which lines continue to be extended (I), and regions where lines continue to be shortened (II), separated by regions where shortening during event 1 is followed by extension during event 2 (III).

(b) Non-coaxial superimposition

additionally produces a region where extension is followed by shortening during event 2 (IV).

FIGURE 4.9 The strain ellipse contains regions in which lines are extended and regions where lines are shortened, which are separated by lines of zero finite elongation.


Transtension and Transpression

Combinations of simple shear (non-coaxial strain) and pure shear (coaxial strain)

FIGURE 4 . 7 A combination of simple shear (a special case of noncoaxial strain) and pure shear (coaxial strain) is called general shear or general non-coaxial strain.

Two types of general shear are transtension (a) and transpression (b), reflecting extension and shortening components.

transtension transpression


Strain quantification


Strain quantities

e = elongation (e1, e2, e3)

s = stretch = (1+e); X, Y, Z

γ = shear strain (ψ is angular shear)

λ = quadratic elongation = (1+e)2 = s2


Strain quantities

Constant area: X:Y = 1

‘

‘

Expressed in elongations

Ratio of angle tangents is strain ratio (and unit circle)


Homework: Strain

A sequence of tilted sandstone beds is unconformably overlain by a unit containing ellipsoidal inclusions (e.g., clasts in a conglomerate). The strain ratio of inclusions in sectional view is X/Y = 4, and dip of underlying beds is 50o. What was angle of dip for beds in sectional view if inclusions were originally spherical?

A diagrammatic brachiopod before (a) and after strain (b). Determine elongation e of the

hinge line and angular shear ψand shear strain γ of the shell.


Natural strain

For vanishingly small strain increment (or infinitesimal strain),

the elongation is:

The natural strain, ε (epsilon), is the summation of these increments:

Integrating gives:


FYI: Mohr construction for strain


Strain states

(a) General strain (X > Y > Z)

(b) Axially symmetric extension

(X > Y = Z)

(c) axially symmetric

shortening (X = Y > Z)

(d) plane strain (X > 1 > Z),

(e) simple shortening (1 > Z).


Representation of strain – map, section

Helvetic “nappes”, Switzerland.


Representation of strain - Flinn and Ramsay diagrams

3D geometry in 2D plot: axial ratios

The Flinn diagram plots the strain

ratios X/Y versus Y/Z.

FIGURE 4.16

The Ramsay diagram uses the logarithm of these ratios; in the case of the natural logarithm (ln) it plots

(ε1 – ε2) versus (ε2 – ε3).

The parameters k and K describe the shape of the strain ellipsoid. Note that volume change produces a parallel shift of the line k = 1 or K = 1.


Finite strain compilation

Plane strain conditions at constant volume (k = 1, ∆ = 0) separate the field of apparent constriction from the field of apparent flattening, which contains most of the strain values.

Plane strain conditions accompanied by 50% volume loss are indicated by the line k = 1, ∆ = –0.5.

FIGURE 4 . 3 0 Compilation of finite strain values from natural markers in a Ramsay diagram that plots log10 X/Y versus log10 Y/Z; the corresponding X/Y and Y/Z ratios are also indicated.

The field inside the dashed line contains an additional 1000 analyses, primarily from slates, that have been omitted for clarification.

Ranges:

�1 < X/Z < 20

�1 < X < 3

�.13 < Z < 1


Strain

FIGURE 4 . 1 Deformed trilobites (Angelina sedgwicki) in a Cambrian slate from Wales. Knowledge about their original symmetry enables us to quantify the strain.


Strain and mechanical contrast

Passive markers have no mechanical contrast: bulk rock strain

Active markers have mechanical contrast: marker strain

FIGURE 4 . 1 8 In two simple experiments we deform a cube of clay with a marble inclusion and a fluid inclusion.

If we require that the elongations in the clay are the same for both runs, the resulting elongations for the marble and fluid bubble are quite different.

This illustrates the different response to stress of materials with mechanical contrast or heterogeneous systems.

A practical example is the determination of strain using (strong) conglomerate clasts in a (weak) clay matrix.


Initially spherical objects

Deformed ooids after

(b) 25% (X/Z = 1.8) shortening

(c) 50% (X/Z = 4.0) shortening

FIGURE 4.19 Ooids (a) that are deformed after (b) 25% (X/Z = 1.8) and (c) 50% (X/Z = 4.0) shortening. The long and short axes of elliptical objects are

measured in three orthogonal sections.

For convenience we assume that these sectional ellipses are parallel to the principal planes. The slope of regression lines through these points (which should intersect the origin) is the strain ratio in that section.

In this example: X/Y = 2, Y/Z = 1.2, and X/Z = 2.4, which gives X/Y/Z = 2.4/1.2/1.

FIGURE 4.20 Strain from initially spherical objects.

FIGURE 4.22 Deformed clasts in a late Paleozoic

conglomerate (Narragansett, RI, USA).


Variably-shaped objects

FIGURE 4 . 2 1 Changes in markers of various shapes under (a) homogeneous strain, (b) coaxial, constant volume strain, and (c) noncoaxial, constant volume strain.

Note especially the change from circle to ellipse and from an initial ellipse to an ellipse of different ratio, and the relative extension and shortening of the markers.


Non-spherical objects Center-to-center method (or Fry method):

spacing varies as function of finite strain

FIGURE 4.23

Strain from

initially spherical

or nonspherical

objects using

the center-to-

center method.

The distance

between object

centers is

measured as a

function of the

angle with a

reference line

A graphical representation of this method using 250 deformed

ooids in a natural sample section

The empty area defines an ellipse with the ratio X/Y; the angle of the long axis with the reference line is also given.


Non-spherical objects Rf/φ method

FIGURE 4.24 Strain from initially spherical or nonspherical objects using the Rf/Φ method.

The ratios of elliptical objects are plotted as a function of their orientation relative to a reference line

(b). Application of this method to 250 ooids (a. same sample as in Figure 4.23) shows the cloud of data points that characterizes this method. From the associated equations we determine the strain ratio and the angle of the long axis with the reference line. The values obtained from using the center-to-center (Figure 4.23) and Rf/Φ (Figure 4.24) methods are

indistinguishable within the resolution of these methods.


Angular change

Breddin method

FIGURE 4.25 Strain is determined from a collection of deformed bilaterally symmetric brachiopods (a), by constructing a graph that correlates the angular shear with the orientation (b). From this (Breddin) graph the orientation of the principal axes and the strain ratio can be determined.


Angular change

FIGURE 4.26 Stretched belemnite from

the Swiss Alps. In longitudinal view (a), calcite filling is seen between the fossil segments, whereas in cross-sectional view (b) the marker remains circular. MN is the location of the sectional view.

FIGURE 4.27 Changes in line length from broken and displaced segments of two once continuous fossils (A and B; belemnites) assuming that the extension direction (and thus ϕ') is known. Simultaneously solving equation λ' =λl' cos2 ϕ' + λ2‘ sin2 ϕ' (the reciprocal of Equation 4.18) for each element gives the principal strains X and Y. An advantage of this method is that volume change (∆) can be directly determined from the relationship X ⋅ Y = l + ∆.


Finite strain compilation

Ranges:

�1 < X/Z < 20

�1 < X < 3

�.13 < Z < 1FIGURE 4.30 Compilation of finite strain values from natural markers in a Ramsay diagram that plots log10 X/Y versus log10 Y/Z; the corresponding X/Y and Y/Z ratios are also indicated.

The field inside the dashed line contains an additional 1000 analyses, primarily from slates, that have been omitted for clarification.

Plane strain conditions at constant volume (k = 1, ∆ = 0) separate the field of apparent constriction from the field of apparent flattening, which contains most of the strain values.

Plane strain conditions accompanied by 50% volume loss are indicated by the line k = 1, ∆ = –0.5.

Dashed area contains additional 1000 analyses not shown


Pure Shear and Simple Shear


Magnitude-orientation plot

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