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. /
Key words:
Image to the upper right of the Orvinfjellet metaconglomerate from the Hecla Hoek basement rocks of Wedel Jarlsberg Land in Svalbard. These clearly are not the shapes these clasts had when they were deposited as a conglomerate. They have been flattened in a plane that dips gently to the right, and stretched out in the direction of the ice axis handle that provides scale here. These are maily carbonate clasts, and in the lower left quadrant one can see how how the different clasts have been molded against each other.
If deformation is finely penetrative (distributed throughout body) at the scale of observation so that lines remain continuous we consider it ductile. Strain distribution can be homogeneous or inhomogeneous (with strain gradients and partitioning). To simplify we will start out considering how to describe and quantify homogeneous strain or distortion. This can then be mapped to document and the strain gradients and patterns associated with inhomogeneous deformation.
Quantitative description is more powerful and permits more rigorous testing of ideas and formulation of models. It allows more precise prediction-retrodiction
There can be a difference between geologic and engineering perspectives 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 and then 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 mechanical character 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. inhomogeneous strain:
Change in length of a line
Shear strain measures the change in angle between lines:
Recognizing elongations and shear strains for a general case of flattening:
Dilation = volume change = (Vf - Vo) / Vo, where Vo is the original volume and Vf is the final volume. 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 that is continually applied, increment after increment.
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 total natural strain for a given point in a progressive straining history is the sum of all these natural strain 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 |
Above is part of an Excel table where the rows represent successive increments of deformation (time proceeding forwards down) and columns represent the various descriptors of strain for the successive increments. Note that the natural strain is constant through this specified history, while the finite elongation is non-linear.
Displacement tensors, strain tensors, finite and infinitesimal.
These three images show how infinitesimal displacement gradient tensors (which describes the displacements gradients of points along a given axis direction) can be envisioned. To the left is the initial configuration of a tiny cube at the origin about to be deformed. The middle diagram shows the deformed state in red and a simple angular shear in the y direction. The tensor rows represent the direction of displacement, and the columns represent the direction of the gradient (what values are influenced by the displacement - in this middle case the z values). Notice that the displacement gradient tensor is not necessarily symmetrical. However, the strain tensor is symmetrical because of how a displacement tensor is transformed into the strain sensor (opposing shear stresses are averaged and the same value is applied to both diagonal locations). The third example with the deformed state in blue shows two elongation displacements, and the associated tensor.
The way you read these displacement gradient tensors is described above.
This diagram shows a simpler 2-D example of the displacement tensor components and how they equate to infinitesimal strain components. All the points that used to be on the z axis have been displaced in the x direction and the displacement gradient in the x direction of points along the z direction is .02. All the points that used to be on the x axis have also been displaced in the z direction but with a different displacement gradient of .01. Note that the lines have changed their angular relationships and hence there are shear strains. Note also that the displacement tensor is not symmetric. By definition the infinitesimal strain gradient is symmetrical. The transformation where the lower left to upper right diagonal displacement gradient components are averaged makes it so. Remember the definition of shear strain, and given the one line is the perpendicular of the other, they have to have the same shear strain.
Link to Excel file that models/visualizes a simple 2-D displacement gradient deformation.
exx exy exz
eyx eyy eyz
ezx ezy ezz
The sign convention for strain is generally similar to stress - if signs are the same then element is positive, if they differ then negative.
This is an easier way to visualize 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.
Examples of strain markers:
Deformed conglomerate from the basement rocks of Spitsbergen. The horizontal scale is roughly 30 cm. There is a mixture of carbonate (e.g. finer grained dark) and silici-clastic clasts (brown, sandy in texture) in this sample. These are deformed to different degrees because of their different mechanical character.
Mixtures of these two strain histories exist - these are two end members.
Start out with pure shear since it is simplest:
Pure shear ccurs during (to a first approximation):
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. You can also download t a program to explore these from Rick Allmendinger's collection of freeware programs for structural geology the one you want is StrainSim.
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.