Lecture index: Intro thoughts on strain. / Deformation = change in material system geometry. / Quantities describing homogeneous strain. / Difference between finite strain and infinitesimal strain. / Infinitesimal strain tensor. / Strain ellipsoid. / Geologic strain markers. / Pure shear and simple shear. /
Readings:
Key words:
If deformation is penetrative (distributed throughout body) at the scale of observation we consider it ductile. Strain distribution can be homogeneous or inhomogenous (with strain gradients and partitioning). To simplify we will start out considering how to describe, quantify homogeneous strain or distortion. This can be mapped to document and see strain gradients and patterns.
Quantitative description is more powerful and permits more rigorous testing of ideas and formulation of models. It allows more precise prediction-retrodiction
Geologic vs. engineering perspective with respect to stress and strain: the engineer is often concerned with predicting the strain from expected forces and internal stresses in order to prevent certain types of strain (those causing system failure). Geologists more often have the strain and want to work back to the original stresses and forces that produced that strain.
An ultimate goal might be to have a model that connects the basic driving forces to the a stress field as it evolves through time to the resulting evolving strain pattern in rocks of initial given mechanical character and structure. Imagine modeling the response of a passive margin as it becomes involved in a continental collision. Stress-strain relationships are mediated by material moduli that describe the strengths of the rock assemblage. Consider that these material moduli are a function of mineralogy, grain size, temperature, pressure and the influence of fluids. The task is enormous, but we can break it down into some of its parts in order to start to understand the system that is the earth's deforming crust. Feedback loops and complex connections exist. As you deform material its material moduli can change making it easier or harder to continue to deform (a process known as strain hardening or strain weakening).
Components of deformation:
In this block diagram of a slump you can see all three components of deformation. The translations is evident by the downward movement of the slump blocks. The rotation is evident as the slope reversal and the backward rotated tree. Such rotation is this particular instance is a direct result of the underlying slip surface being curved. Movement on listric faults produces a block rotation. The transverse ridges and transverse cracks involve internal distortion.
Diagram source: http://geology.wr.usgs.gov/wgmt/elnino/deserten/processes.html
Homogeneous vs. inhomogenous strain:
Change in length of a line
Shear strain measures the change in angle between lines:
We can use the diagram above to explore these measures of strain. An initial square, circle and lines will mark the deformation.
Dilation = volume change = (Vd - Vu) / Vu. Note that this indicates nothing about accompanying geometry of volume change, just the magnitude. Geologic processes involving dilation:
Finite strain compares only the initial and final states. Remember, there is an infinite variety of ways to get from the initial to the former state. What if you want to model a strain history? Then you can consider a small increment of strain, continually applied and you need somewhat different equations.
An increment of natural strain ei = dl / l' , where dl is the change in length and l' is length at beginning of increment, but not the original length, since the previous strain increment changed l'. The natural strain is the sum of all the increments. If you integrate from an initial length to a final length, then by definition you end up with the natural strain = ln ( final length/ original length).
Substituting the fact that stretch = S = final
length/ original length = (1 + e), where e is the finite strain
equivalent, then the natural strain = ln ( 1+finite strain).
example: if ei = .1 and initial length lo = 1
| strain increment | length of line | finite elongation | natural strain |
| 0.1 | 11 | 0.1 | 0.09531018 |
| 0.1 | 12.1 | 0.21 | 0.19062036 |
| 0.1 | 13.31 | 0.331 | 0.28593054 |
| 0.1 | 14.641 | 0.4641 | 0.38124072 |
| 0.1 | 16.105 | 0.61051 | 0.4765509 |
| 0.1 | 17.716 | 0.77156 | 0.57186108 |
| 0.1 | 19.487 | 0.94872 | 0.66717126 |
| 0.1 | 21.436 | 1.14359 | 0.76248144 |
| 0.1 | 23.579 | 1.35795 | 0.85779162 |
This is similar in form to stress tensor, 3-D
description of homogeneous strain
eij
subscripts: i is direction of displacement, j is coordinate axis point original on.
For small angles eij is approximately tan of shear strain angle, i.e.
it is the shear strain.
exx exy exz
eyx eyy eyz
ezx ezy ezz
Again the one diagonal is composed of elongations, the rest are shear strains.
One sign convention for strain is similar to stress - if signs are same then element is positive, if they differ then negative
This is an easier way to envision strain. Given an initial sphere, what shape does it have after deformation? Ellipse axes are principal strains.
Various ellipsoid shapes:
Strains of lines other than principal strain directions (see handout)
Formulas for infinitesimal strain as function of angle from e1
e = (e1 + e2 / 2) + (e1 - e2 / 2) cos 2ø
¥ = shear strain = (e1 - e2) sin 2ø
If you compare to formulas for Mohr diagram then see can find strain of any line using the Mohr circle construct. The y axes however must be ¥/2 instead of just shear strain. Link to use of Mohr circle for plane strain.
With advent of computers it is in many ways easier just to use them to compute values. You could develop an excel spreadsheet that would compute the strains on any line at an angle to principal strain axes.
Adjacent
is an image of jasperized Carboniferous crinoids in a low grade
marble from a major Tertiary shear zone in western Spitsbergen.
Careful inspection shows how they are deformed into natural ellipses.
The low-angle diagonal subvertical surface is a solution cleavage,
while the concentration of crinoid fragments is on a sub-horizontal
bedding surface. Since the yellow carbonate matrix and
the red jasperized crinoids are composed of different minerals
they have different strengths, with the matrix being weaker. The
crinoid ellipses thus represent a minimum strain the bulk rock
has seen. A tip of a swiss army knife blade is in the lower center
for scale.
Examples of strain markers:
Mixtures of these two strain histories exist - these are two end members.
Start out with pure shear since it is simplest:
Occurs during:
Simple shear deformation path best modeled by incremental slip of a stack of cards. Note that:
Simple shear occurs during:
We have computer program in the RSAL lab that model these different behaviors, separately and in combo. You should play with them.
On to stress-strain relationships and rheology.
Copyright Harmon D. Maher Jr., This may be used for non-profit educational purposes as long as proper attribution is given. Otherwise, please contact me. Thank you.