DEFORMATION                            

Deformation- produced in response to Stress

    Depends upon:   

            Type of stress applied

        Rock properties (minerals, discontinuities, etc)

        Temperature

        Depth

        Time

Deformation= change in position, shape or volume or rotation as a result of applied stress.

Describes the complete displacement field of a set of points in a body relative to an external reference frame.

4 deformation components are:

  1. Translation- movement from initial location.

  2. Rotation- spin about an axis

  3. Distortion- change in shape (Strain)- describes displacement field of points within the body; i.e., from an internal reference frame.

   4.Dilation- volume change

Strain Axes:

     X= maximum direction of extension

                    (or minimal compressive strain

     Y= intermediate strain axis

     Z= maximum direction of shortening

                    (or minimum extension).

Relationships between Stress and Strain:

Since strain results from the actions of stresses, a geometrical relationship between the two must exist. Documenting this relationship is difficult at best. Do not assume the stress axes ₁, _{2, 3} correspond with strain axes X, Y and Z.

Knowledge of Undeformed States

Strain analysis requires a knowledge of the original undeformed state of the material (rare in nature).

Homogeneous Strain- Situation in which strain in all points of a rock body is the same

        Original straight lines remain straight

    Original parallel lines remain parallel

    Circles become ellipses;

        3-D spheres become ellipsoids

Material Lines- lines that contain recognizable features (e.g., grains, fossils) that do not rotate relative to one another during deformation, although the length of the lines does change.

          2-Dimensional Homogeneous Strain:

two orientations of material lines remain perpendicular before and after strain, defining the Strain Ellipse

          3-Dimensional Homogeneous Strain:

3 or more material lines remain perpendicular before and after strain, defining the Principal Strain Axes X > Y > Z of the Strain Ellipsoid

Heterogeneous Strain- Strain is different in various parts of the rock body.

1 or more of homogeneous strain conditions do not apply   

        Original straight lines do not remain straight

    Original parallel lines do not remain parallel

    Circles do not become ellipses;

        3-D spheres do not become ellipsoids.

All strains may be expressed in terms of extensions and rotations so that a knowledge of e, l or w are sufficient to describe the strain history for any homogeneous domain.

STRAIN KINEMATICS

Strain Path- Kinematic strain development;

    describes a series of incremental strain events  

   cumulatively resulting in a finite strain state.

Incremental Strain-

        Intermediate strain steps

       describe separate strain conditions;

        usually difficult to ascertain.

Finite Strain-

Measure of the strain from an initial to final state

 Represents the sum of the incremental strains.

Coaxial vs. Non Coaxial Strain

Coaxial Strain- No rotation of the incremental strain axes from an initial to final strain state.

The same Material Lines remain the principal strain axes throughout deformation

No Rotation of Material Lines (Zero Internal Vorticity)= Pure Shear

  Pure Shear (Irrotational Strain)-

X, Y, Z axes do not rotate during progressive strain.

   Uniform elongation in 1 direction

   Uniform contraction in perpendicular direction

   Principal strain axes correspond to principle stress axes throughout deformation

   Pure shear is a coaxial strain with no change in volume (rigid, i.e. unrealistic?, interpretation)

   Strain axes are parallel to principal stress axes:

   ₁= Z    ₂=Y    ₃=X

Non-Coaxial Strain  (Rotational Strain)-

   Axes of strain ellipsoid rotate through time

During incremental strain steps, the principal strain axes do not remain the same

Principal strain axes (Material Lines) occur throughout incremental strain events; however, different principal strain axes occur at each incremental step.

(Card Deck Analogy)

Simple Shear (Rotational Shear)- Non coaxial strain in which the distance perpendicular to the shear plane remains constant (thickness of card deck remains the same)

All points move parallel to a fixed direction with an amount of displacement proportional to a distance from some defined plane (e.g., parallel to face of cards)

Strain axes do not remain parallel during progressive deformation

Axes of strain ellipsoid rotate through time. X, Y and Z axes rotate during progressive deformation for a fixed single stress orientation.

Therefore, strain axes do not remain parallel during deformation.

Hence, the direction of maximum elongation is not parallel to the direction of minimum compressive stress or maximum tensional stress.

The direction of maximum shortening (minimum extension) is not parallel to the direction of minimum tension or maximum compressive stress.

General Shear- combination of pure shear and simple shear; common.

MEASURING DEFORMATION

Length Changes

Volume Changes

Angular (Rotational) Changes

1. LENGTH DEFORMATION

Longitudinal Strain (e) = extension   

Length Change (e)= Change in Length     e=L-L_o       

                                  Original Length             L_o

                            _{e (extension) =} (L-L_o)/ L_o

L_o=Original Length  

L= Final Length           

e (Extension) is a dimensionless quantity            

Shortening- negative values         e<0

 Extension- positive values           e>0

2. Quadratic Elongation ()         

   alternative expression for length changes

   or

= (l+e) ².        

3. Stretch (s)          s= ()^1/2

Quadratic Equation and Stretch values are useful in describing the lengths of the principal axes of the strain ellipsoid:

 X² = l₁          Y² = l₂          Z² = l₃

 X=s₁          Y= s₂          Z= s₃

VOLUME DEFORMATION

Volumetric Strain ()    

Volume Change ()= (V-V₀)/V₀     

        V₀= Original Volume

          V= Final Volume         

          is a dimensionless quantity

        Decrease volume- negative values

          Increase volume- positive values

ROTATIONAL DEFORMATION

Angular Strain()  Rotational Change

Change in angle between 2 initially perpendicular lines

Shear Strain and Angular Shear are dimensionless.

Shear Strain ()= tan .          

= angular shear  (psi)-

     = deflection from an originally right angle

  = shear strain (gamma)

- change in angle between 2 lines

Strain in 2 Dimensions- consider a circle of unit radius (r=1) and a center �O� that has been deformed into an ellipse with a major (maximum elongation) axis (₁) and a minor (minimum elongation) axis (₂). Such an ellipse is called a �strain ellipse�. Any point �P� on an original circle with coordinates X, Z has moved to a new position with coordinates X�, Z�. Note that has also changed as a result of elliptical strain to �. Note also that the length of line O-P has been changed to O-P�.

Strain in 3 Dimensions: Strain Ellipsoid with axes X, Y and Z.

Check out the following Webpage for Stress and Strain visualization: http://www.geology.sdsu.edu/visualstructure/vss/htm_hlp/index.htm

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