Stress, Strain and Their Relationship

Marianne Casamatis, Laura Matarese, Danielle Sharpe and Katherine Zunica



Table of contents


PART 1

1) Introduction


PART 2 - STRESS

2) What is stress?

                            - force and traction


3) Stress acting on a plane

                            - Mohr diagram


4) Stress acting on a point

                           - the stress ellipsoid

                           - properties of the stress ellipsoid
                                             - principal stress axes

                           - the stress tensor

PART 3 - STRAIN


5) What is strain?


6) Types of Strain : Homogeneous vs. Heterogeneous


7)  The Strain Ellipsoid


8)  The Flinn Diagram


9)  The Finite Strain Ellipsoid


10)  Strain Path
                          - Coaxial
 
                          - Non co-axial

11)  Incremental and Finite Strain
                           - The incremental strain ellipsoid
           
12) Strain regime : pure shear vs. simple shear


  
PART 4 - THE STRESS/STRAIN RELATIONSHIP


13) Indications of stress and strain.


14)  Small-scale strain features in the tectonic regime.


15) Pure and Simple Shear in nature


16) Incremental strain and Kinematics analysis


17) Stress Perturbation


PART 5 - Conclusion, References and Glossary

Introduction

This report gives an overview of two important concepts within structural geology; stress and strain. These concepts are important because they allow us, to some extent,  formulate and describe the deformation history on both a micro and macro scale.

The report will firstly outline how stress and strain are defined and then how the two concepts are related to one another. 

Stress can be defined in terms of force and traction. Stress can be divided into two categories; stress on a plane and stress at a point. Stress acting on a plane consists of two components; normal and shear stress. Stress acting on a plane can be represented and calculated using the Mohr diagram for stress (also outlined in the first section of the report). Stress on a point can be visualised using the stress ellipsoid. The construction and components of the stress ellipsoid, including the principal stress axes, will be outlined in the first section of the report. Stress at a point can also be defined mathemetically, using the stress tensor

The second section of the report will deal with strain. It will outline what strain is and how it can be represented visually in terms of the strain ellipsoid. There will be a discussion of the two types of strain; homogeneous and heterogeneous. The Flinn diagram is important to the representation of strain. The Flinn diagram is a graph which shows all possible configurations of the strain ellipsoid in terms of the ratio of its axes, strain intensity and shape of the strain ellipsoid.  We will discuss the strain path, which describes the intermediate stages during the process of straining, and this will include co-axial and non co-axial strain paths.  We will then discuss incremental strain, and how it leads us to the Finite Strain Ellipsoid, then we will relate this to pure and simple shear.

The final section of the report will discuss the nature of the relationship between stress and strain. It will also look at ways of how we can derive the orientation of the stress and finite strain ellipsoids from field observations. 




STRESS


What is Stress?

Before we look at defining stress, we need to define the terms FORCE and TRACTION.

What is force?

A force changes the velocity of an object or the state of rest/motion of an object.( http://www.gesc.ttu.edu/yoshinobu/Yosh_Struc_Folder )

There are two types of forces: body and surface forces. "Forces that result from action of a field at every point within the body are called body forces." (van der Pluijm, Marshak, 1997 p37). Forces that act on a specific area of the body are surface forces. (van der Pluijm et al, 1997). This report will deal with surface forces as they are the most important type of force to structural geology. (However, gravity is a body force that is always present on Earth.).

Force is measured in Newtons per metre squared.  (N/m^2)

What is Traction?

A traction is a force vector. The fact that it is a vector then implies that is has both magnitude and direction. The magnitude is the length of the vector and the way it is pointed is its direction. In terms of traction, stress is two tractions with the same magnitude but have opposing directions. "Stress really refers to a whole collection of tractions acting on a single point at any given time."( http://www.gesc.ttu.edu/yoshinobu/Yosh_Struc_Folder ).


Caption for figure below...
Caption for figure above

So, what is stress?

Stress can be calculated mathematically using this equation: Force/Area.

So, we can think of stress in terms of the intensity of force for a given area or a measure of how concentrated the force is.

 Stress is usually measured in megapascals. (MPa).  (1 N/m^2=1Pa)

Stress can be thought of as acting on a surface or a point. Stress on a surface and a point will now be discussed. 




Stress acting on a Plane

In this section, we are examining stress acting on a single plane, and thus we are looking at stress in 2-dimensions.

As previously mentioned, stress is a vector. Stress can be acting in any orientation to a plane. Stress acting on a single plane can be reduced to two vector components (Pluijm etal, 1997). These are the normal stress component and the shear stress component. On either side of the plane, these components have opposing tractions



The normal stress component has the symbol  sigma n. It is the stress component perpendicular to the plane.

The shear stress component has the symbol Tau or sigma  s. It is the stress component that is parallel to the axes of the plane. This will be shown in more detail, later in the report.

The Mohr Diagram for Stress

The Mohr circle is a useful tool in structural geology. It is a graphical way to represent and calculate stress acting on a plane. The normal and shear stress components are calculated using the fundamental stress equations below. The diameter of the mohr circle is the differential stress (sigma 1 - sigma 3). The vertical axis of the graph shows the magnitude of the shear stress component and the horizontal axis shows the magnitude of the normal stress component. Sigma 1 and sigma 3 lie on the horizontal axis as they are orthoganol to the plane and therefore contain no amount of shear stress.

The diameter of the circle is indicative of the material's strength. The larger the diameter of the circle, the larger the strength of the material under stress.

Theta in the diagram is the angle between the normal stress vector and the orientation of sigma 1.




www.geology.sdsu.edu/visualstruture/vss/htm_hlp/mor_cir.htm
The diagram above shows the mohr circle
Fundamental Stress equation for normal stress.
Fundamental Stress equation for shear stress


Stress at a Point


a) The stress ellipsoid

A point defines the intersection of an infinite number of planes, each with a different orientation (Pluijm etal, 1997). So, the state of stress acting on a point describes all the stresses acting on these planes.

Each of these planes will have the normal and shear stress components that have been described above. The magnitudes of these components vary as a function of the orientation of the plane. This property can be illustrated in three- dimensions to obtain the stress ellipsoid.

Visually, the length of the line of the stress vector is representative of the magnitude of the stress applied. The length of the stress vector will vary due to the orientation of the plane. Imagine this property being applied to the infinite number of planes that a point intersects. This report will not comment on how the lengths of the stress vectors are determined, only on what outcome we can derive from them.

               


If we envelope these stress vectors, we obtain an ellipsoid. This is the stress ellipsoid. It fully describes the state of stress at a point (Pluijm etal, 1997).



The ellipsoid allows us to find the stress for any plane.

b) Properties of the Stress Ellipsoid

An ellipsoid is defined by three axes.

Caption...


In the stress ellipsoid, these axes are defined as the principal stresses. These three principal stress axes are orthogonal to one another. They are also perpendicular to three plane

These planes have the only orientation where the shear stresses are equal to zero. Thus the planes only have the normal stress acting on them (lec7 net reference). These three planes are called the principal planes of stress. However, in this report we are mainly concerned with the axes of the ellipsoid.

The principal stresses are vectors. That is, they have magnitude and direction.

We can describe the state of stress of a body by specifying the orientation and magnitude of these axes.


c) The Principal Stress axes

The principal stress axes of the stress ellipsoid are labelled sigma1, sigma2 and sigma3



1 is the principal stress axis (the direction of maximum stress)

2 is the intermediate principal stress axis.

3 is the least principal stress axis (the direction of minimum stress)

Or, in mathematical terms, sigma1>sigma2>sigma3.


There are several common states of stress that can be defined by the relationships of the principal stresses. These stress states are isotropic, anisotropic and deviatoric. Types of anisotropic states of stress are uniaxial, biaxial and triaxial.


An isotropic state of stress is where all three principal stresses are equal in magnitude. The ellipsoid is actually a sphere in this case.

sigma1=sigma2=sigma3



Anisotropic is a stress state where at least one axis has a different magnitude to the other axes. This describes an ellipsoid



Deviatoric stress is the part of the total stress that is left after the mean stress is removed. Deviatoric stress is equivalent to tectonic stress and is the sress responsible for deformation.

Mean stress, sigma m, is given by sigma m=(sigma1+sigma2+sigma3)/3. Deviatoric stress has the symbol sigma dev.

So,

sigma total=sigma m + sigma dev




Uniaxial stress can be in tension or compression. The sign convention for tension in geology is negative, and compression is positive (Pluijm etal, 1997). The ellipsoid is 'needle-like'.

Uniaxial tension: sigma1=sigma2=0; sigma3<0 (Pluijm etal, 1997).

Uniaxial compression: sigma2= sigma3= 0; sigma1>0.

Uniaxial compression


Biaxial stress is where one axis equals zero. For example, sigma 1>0>sigma 3

The general triaxial state of stress is where none of the three principle stress axis can be zero. That is,

sigma 1>sigma 2>sigma 3=/=0


The Stress Tensor

The Stress Ellipsoid is one description for the state of stress at a point. There is also a mathematical description that defines this state of stress, called the Stress Tensor. We will briefly detail this mode of defining stress. For a more detailed explanation refer to the reference list.

The orientation and magnitude of the state of stress of a body can be defined in terms of its components in a specific Cartesian reference frame. A Cartesian reference frame has three mutually perpendicular coordinate axes, X, Y, and Z. Or in this image X1, X2 and X3.




The image shows a stress acting on a plane (X1, X2). Its vector components are also shown on this image. The normal stress component is parallel to X3. The shear stress components are expressed as being parallel to X1 and X2.

In three dimensions, we can think of the point of stress as a cube.




The stress acting on the three faces perpendicular to the axes of the cube can be resolved into their component parts. For example, the face normal to the X1 axis can be resolved into its normal component, sigma11 and the shear components are sigma12 and sigma13. This can be done for the face with the normal component sigma22 and sigma33. This gives us a total of nine stress components. This is given in the form of a matrix:

                                  In the direction of:
                                  X1              X2            X3



We can relate the stress tensor cube to the stress ellipsoid. For example, the stress ellipsoid has three principal planes of stress, where they contain no shear stress components. We can consider this state on the cube.




The plane (X1, X2) can be oriented so there are no shear stress components and the normal components, 1, 2, 3 are left acting on these planes. These normal components represent the principal stresses. The matrix would be reduced to:



Again, this can be visualised as the stress ellipsoid









Strain



What is Strain?

Strain is the response of rocks to stress. It describes the final shape of a rock in terms of the initial shape.

During deformation, a rock will change in size and shape. Deformation describes the complete transformation from the initial to the final geometry of a body, and can be broken down into 4 main components:

  • Translation (movement from one place to another)

  • Rotation (spin around an axis)

  • Distortion (change in shape)

  • Dilation (volume change)

Strain specifically relates to the changes of points in a body relative to each other.      

From this we can see that strain relates to the distortion component of deformation.

Types of Strain: Homogeneous vs. Heterogeneous


There are two types of strain: Homogeneous and Heterogeneous.

The way we differentiate between the two is by examining marker lines drawn on an object both before and after deformation.

Straight marker lines remain straight after homogeneous deformation.

Straight marker lines become curved after heterogenous deformation.



Homogeneous Strain


Homogeneous strain results when any two portions of a body which were similar in form and orientation before strain, are still similar in form and orientation after the strain.

As a consequence, straight lines remain straight, parallel lines remain parallel, and planar surfaces remain planar.

In two dimensions, circles will become ellipses, and in three dimensions spheres will become ellipsoids


Caption...


Heterogeneous Strain

Strain is heterogeneous when it varies across the surface of an object.

Changes in the size and shape of small parts of the body are proportionately different from place to place.

As a consequence, straight lines become curved, planes become curved surfaces, and parallel lines generally do not remain parallel after deformation.

This implies the presence of a strain gradient.


This is represented in the diagram below. ****


Captions......

In  order to analyse a heterogeneously strained body, we need to break it down into homogeneous portions. We therefore divide a deformed body into volumes that are small enough for the deformation to be described as locally homogeneous.


Captions....


This can be seen in figure ******** above. The picture on the left represents the undeformed state. The picture on the right depicts heterogeneous strain on the scale of the whole block. We divide the whole block into small sections, as depicted by the yellow circles. At the scale of each individual cell containing a yellow circle, the circles have been transformed into ellipses, and the strain at each cell level is now homogeneous.


The Strain Ellipsoid


The strain ellipsoid is used as a three dimensional way to represent strain.

The strain ellipsoid results from the homogeneous deformation of an imaginary sphere, which represents the undeformed state of a body.

In any homogeneously strained, three dimensional body, there will be at least three lines of particles, also known as material lines, that will not rotate relative to each other. After strain, we will therefore have three material lines that remain perpendicular. These lines define the axes of an ellipsoid, and are known as the principal strain axes. They are referred to as X, Y and Z, where


                             X > Y > Z


             X = maximum direction of extension

             Y = intermediate strain axis

             Z = maximum direction of shortening


These axes are also known as lamda 1,2 and 3, where
                                          lamda 1 >  lamda 2 >  lamda 3


The axes of the strain ellipsoid will therefore be different in length to the length of the axes in an undeformed sphere. This difference is a measure of the strain magnitude.

The Flinn Diagram

 
 
Figure number with caption............

Figure number..... with caption..... Patrice's lecture slides, perhaps put a link to module 2........

The Flinn diagram shows all  the configurations that the strain ellipsoid can take. The parameters which influence the character of the stress ellipsoid are K, D and two ratios of the principal strain ellipsoid axes. (as shown in the diagram above).

The parameter K characterises the shape of the strain ellipsoid and is found using the equation given above. on the diagram K relates to the slope of the line. Ellipsoids that are close to a pancake shape (when K is close to 0), are found near the horizontal axis. Ellipsoids resembling a cigar shape are found when K approaches infinity, near the vertical axis.

The parameter, D,  refers to the strain intensity and is given by the equation in the above figure on the right. From the Flinn diagram one can ascertain the strain intensity by how far away from the origin the ellipsoid lies. Different strain intensities are shown in the figure to the left by small, red squares.

The horizontal and vertical axes are simple ratios between the principal strain axes of the stress ellipsoid. The vertical axis: lambda1/lambda2, the horizontal axis: lambda2/lambda3.

From field observations of structures such as  foliation planes and stretching lineations, we can derive the orientation of the principal axes of the strain ellipsoid. The foliation plane lies parallel to the plane of lambda 1 and lambda 2 and the stretching lineation lies parallel to lambda 1.

NB: We use a logarithmic scale so that the ellipsoids are spread out. Sometimes the ratios between the principal axes are quite large so it is helpful to use a logarithmic scale.





The Finite Strain Ellipsoid


The finite strain ellipsoid (FSE) represents the end product of the homogeneous deformation of an imaginary sphere.

It does not give us any information about the strain history, or the mode of shear, that is, whether the shear was pure or simple.


For this we need to know about the strain path.



The Strain Path


A rock does not undergo ductile deformation instantaneously. The strain states that a rock will progressively pass through, to reach its final deformed state, is known as the strain path.


In nature, we generally see rocks in their final strained state, and we must deduce the initial undeformed state. The strain path gives us a description of the intermediate stages during the process of straining.

Strain paths can be either coaxial or non-coaxial.


Coaxial strain path

A co-axial strain path is where the strain axes are parallel to the same material lines throughout the straining.

Therefore, the principal axes of the strain ellipsoid do not rotate during deformation.

An example of this is pure shear.


Non-coaxial

A non-coaxial strain path is where the strain axes are parallel to different material lines during each infinitesimal increment of straining.

Therefore, the principal axes of the strain ellipsoid move through the material during deformation. At any given increment of strain, the principal axes lie in a different physical part of the deforming material.

An example of this is simple shear.




Incremental and Finite Strain


Most of the structures we see in nature reflect the total strain. The total strain is also referred to as the finite strain. It is independent of the intermediate steps. Finite strain can be the same for two rocks having different strain paths.

Incremental strain is the increment of distortion whose sum led to finite strain.

The finite strain is the sum of all of the incremental strains.

Incremental strain is represented by the Incremental Strain Ellipsoid (ISE).

The axes of the ISE do not change in orientation or magnitude from one increment to the next.

Figure ****** below represents the increments of strain during pure shear.

The incremental strain ellipse is at the bottom and does not change.




Strain Regime: Pure Shear vs. Simple Shear

As already seen the FSE does not give us any information regarding the strain history.

The FSE can develop according to the two end members of strain regime: pure shear and simple shear


During pure shear, the principal axes of the FSE do not rotate during deformation.

The axes of the FSE remain parallel to that of the ISE.

During simple shear, the magnitude and orientation of the axes of the FSE will change. The axes of the FSE will rotate during deformation, and will not remain parallel to the axes of the ISE. The rotation of the axes will give us information relating to the kinematic of the deformation.

Often in nature both strain regimes are present with one being dominant over the other and therefore dominating the appearance of foliation, shear zones and features such as clasts and rigid objects. Together they are called general shear,  with symmetric patterns indicating dominant pure shear and asymmetric patterns indicating dominant simple shear.


Figure whatever.....


The Relationship Between Stress and Strain.


We have seen that the process of deformation can be defined in terms of stress and strain. The relationship between these is fairly basic, ‘strain is a distortion as a result of applied stress’ (Chapman 1995). In other words stress (δ 1, δ2, δ3) is the force acting on a point (or plane) before or during distortion. Strain (λ1, λ2, λ3) is the distortion itself.

The strain path, paleostress or history of deformation can therefore sometimes be extrapolated by looking at the end product. The Finite Strain Ellipsoid (FSE) as explained before is a graphical 3D model of this end product. Studying the FSE or Finite Strain Analysis (FSA) examines such things as stress, strain, strain regime, incremental strain and kinematics. Actual physical features are used to explore these ideas, and the distortion with time examined.

Indications of stress and strain.

All the theoretical models so far discussed have the undeformed state represented as a perfect sphere with all stresses being equal. This is in fact impossible to find in nature, but a few features do come close. Ooides in limestone, conglomerate stones (river worn), vesicles and amygdules in basalt flows and reduction zones in shales are all examples of naturally occuring features which approximate a sphere(van Pluijn). Depositational conditions must be taken into account and any process which can deform the sphere other than stress and strain. Basically anything with a set known shape before deformation can be used as long as it is homogenous with its groundmass, meaning they have the same approximate densitys and internal strength. Garnet clasts in mica shists are not. They tend to disassociate and remineralise with pressure and form pressure shadows or windows, they dont deform homogenously within themselves.
These then in pure shear form pinched in sections on either side of the clast and are symmetric. If the clast is rotating during or after creation of the pressure shadow the pinched edges curve giving an asymmetric shape.






Small-scale strain features in a tectonic regime.


Paleostress analysis uses the relationship between stress and strain to examine the deformation history. This for example is a block in compression. From the strain features present, the vertical styolitic joints, horizontal extensional fractures and the en echèlon features some idea of the stress acting upon the unit can be deduced. En echèlon features can be an indication of incremental strain as new ones open up as a result of strain over time.


Pure and Simple shear in nature.

As stated previously, general shear is more commonly found with either pure or simple shear being the dominant regime.

The pictures below show, on the left, a dominant simple shear regime, and on the right, a dominant pure shear. Note the asymmetry of the left and symmetry on the right.





Incremental strain and Kinematics analysis.

The ISE and the Incremental Strain Analysis (ISA) that can be done from this are inextricably linked to the strain regime, as incremental strain is the change of strain through time.
ISA is basically the relationship between the stress axis and strain axis, and both axes have to be parallel to make a comparison. Pure shear, which doesn’t rotate with time, can be analyzed. Simple shear cannot be analysed, as while the stress axis remains constant with time, the strain axis rotates. This rotation gives the kinematics of the deformation also known as internal vorticity which is measured by its degree of non co-axiality or the degree it has deviated from the original parallel relationship between δ1// λ3, δ3//λ1 and δ2// λ2.

Therefore Internal vorticity = Wk (kinematics vorticity number) and α = degree of deviation.


Then:                                 Wk = Cos. α


When:             Wk =0     pure shear.

                  0>Wk>1         general shear.

                                        Wk =1            simple shear.


Stress perturbation.


Stress perturbation is a term used to describe the behavior of a stress field once it looses its homogeneity. 



logically if enough stress is placed on an area strain and distortion follows causing extensional fractures (an array of these is called en echèlon) with dominant simple shear. If the stress/strain force overwhelms the mechanical strength of an area a mechanical discontinuity/instability will occur, commonly called faults. *** this illustration shows one such scenario where an extensional fracture with dominant simple shear has been dissected and driven apart by a transform fracture/fault. This has the effect of mechanically isolating the total into two homogenous(isotropic) sections either side of the failure. These two sections are continuing to rotate under the stresses δ1 and δ3, that deformed the extensional fracture. But since they are now separate the bottom right section will continue in a clockwise direction and the upper left in an anti-clockwise.



Conclusion



References



Glossary

Material Lines
:  Lines that contain recognisable features, (for example, grains or fossils), that do not rotate relative to one another during deformation.