Structural Geology
Lecture 3
Mapping of Points and Displacements during Deformation
(introduction to kinematic analysis)

In this lecture techniques for kinematic analysis are introduced.  The final product is a full accounting for rock strain which is at the heart of kinematic analysis.  We can start our analysis with a one-dimensional example of strain.  Strain ( e ) is a dimensionless quantity defined as a change in length (Dl) per unit length (l).
	e  =  Dl /l                                                                        
Strain is actually defined near a point by the limiting process of differential calculus.

	e  =  lim ( Dl /l)
	       l Þ 0                                                                          

Now differential quantities can be introduced by letting dx = l and du = Dl

	                     du               ¶u
	e  =   lim        ___      =      ___
	       dx Þ 0    dx                ¶x                                                  

Strain in rocks is mapped by following two points separated by Dx through a deformation (Fig 3-1).  One dimensional strain may be illustrated by drawing a line out from the origin (O) through two points such as P and Q.

  	Dx  +  Du  =  Q'  -  P'

(Fig. 3-1)

If the line is stretched the points P and Q go to P' and Q', respectively.   The length from the origin to P is denoted as x whereas the distance between P and Q is Dx.  The stretch from P to P' is u so that the total distance from P' to the origin is x + u.  The stretch of Dx is Du so that the distance between P' and Q' is Dx + Du.  The strain of PQ is then 

	e    =    Dl/l    =    Du/Dx                                                            

If  Du = 0.1 and Dx = 1, then e  =   .1/1  =  10%.  
Deformation is conveniently separated into three components, of which two are displayed in this one-dimensional analysis.  In one dimension there can only be a rigid-body translation and a stretch.  The third component shows up in two and three dimensions where there is an additional deformation knows as a rigid rotation were the body spins about an axis.  These concepts will be more fully developed as we progress through the elements of strain analysis.

Motion of a Point in a Deformed Body

Figure 3-1 demonstrates that for strain analysis we must keep track of the relative position of two points within a deforming body.  However, before looking at the relative motion of two points, it is instructive to consider the absolute motion of one point. Deformation mapping involves the tracing of a material point within a body from its undeformed position to its deformed position.   The material point in its undeformed position is specified by a vector X with components X1, X2, X3 or Xi.  In the new or deformed position the vector x has components x1, x2, x3 or xi  (Fig 3-2).  Note that in Figure 3-2 the undeformed body is shown as a circle, a very common geological shape.  In the deformed state that body becomes an ellipse.  The point that we are watching is the center of the circle.

NOTE:  You will use capital letters for coordinates and coordinate axes in the unreformed state and lower-case letters for coordinates and coordinate axes in the deformed state.  This convention is used largely to conform with Means (1976).

(Fig. 3-2:  Until this figure is redrawn xi = X’i)

There two techniques for keeping track of what happened to the center of the undeformed body (i.e., the circle).  First, we can develop a series of equations relating the final resting spot of the center of the circle.  Or, we can develop a series of equations describing a vector pointing to the final resting spot from the initial position of the center of the circle.  

Mapping of Points

One way of looking at the description of the 2-dimensional motion of a particle of rock is through the following functional relationship which is described by the deformation equations.  These equations act to locate the final resting place of a point in a deformed body using the location of the point before deformation.
	
	x1  =  f(X1, X2)

	x2  =  f(X1, X2)                                                                      
or
	xi  =  f(Xj)                                                                              

In three dimensions this same relationship is  
	
	x1  =  f(X1, X2, X3)
	
	x2  =  f(X1, X2, X3)
	
	x3  =  f(X1, X2, X3)
	                                                                     
or
	xi  =  f(Xj)  

Written in their complete form, the deformation equations are as follows:

	x1  =  a0  +  a1.X1  +  a2.X2  +  a3.X3
	
	x2  =  b0  +  b1.X1  +  b2.X2  +  a3.X3
	
	x3  =  c0  +  c1.X1  +  c2.X2  +  c3.X3

The constants, a0, b0, and c0 specify the rigid-body translation. If there is no rigid-body translation, then the deformation equations for a homogeneous deformation may be written in matrix form:

Later we will use a double subscripted variable to represent the nine components of the deformation equations.

There are some special cases for deformation which are important in simplifying geological analysis.  In three dimensions one example of a homogeneous deformation can be represented by the following equations:
	
	x1  =  a0  +  a1.X1  +  a2.X2
	
	x2  =  b0  +  b1.X1  +  b2.X2
	
	x3  =  X3                                                                            

The reason that this is called a homogeneous deformation is that all points x'i are linearly related to points xi.  This is an also example of deformation in plane strain where all motion is parallel to the plane normal to the x3 axis.  (Plane strain is further discussed with regard to faulting in a future lecture).  The point (0,0,0) goes to the point (a0,b0,0) which defines the rigid body translation.   Points along the direction x1 where x2 = 0 go to the point
	
	x1  =  a0  +  a1.X1      
		
	x2  =  b0  +  b1.X1                                                             

In contrast a non-homogeneous deformation the points x'i are related to xi in a nonlinear manner.  The following is one example of such a deformation in plane strain
	
	x1  =  a0  +  a1.X1  +  a2.X2  +  a3X12
	
	x2  =  b0  +  b1.X1  +  b2.X2  +  b3X12
	
	x3  =  X3 

We can introduce an example of mapping of a deformation using the following matrix to specify our deformation equations:

The three equations mapping the deformation might be the following provided it is understood that the rigid body translation (a0 = 5) is not specified by the matrix given above (Fig. 3-3):
	
	x1  =  5  +  X1  -  X2
	
	x2  =        2X1  +  2X2
	
	x3  =  X3 
                                            
Figure 3-3 is a mapping of three points of an undeformed body to its deformed shape.  In this example you can think of the undeformed body as the outline of a shark’s tooth.  You will see that the deformation actually ‘sharpened’ the shark’s tooth.  All of this was done using the deformation equations as given above. 

(Fig 3-3)

Displacement Vector
 
The motion of the material point from Xi to xi is also described by a displacement uo or (uo)i.  uo ,which is a displacement vector, is nothing more than the final position, x, minus the initial position, X.  In Figure 3-3 the motion of the corner’s of the shark’s tooth was shown by two vectors.  The vectors may be expressed in vector components
	
	(u0)1   =    x1   -   X1                                                             
and	
	(u0)2   =    x2   -   X2                                                          
or
	(uo)i   =   xi   -  Xi                                                                
or 
	u0  =   x  -  X                                                                

These equations are known as the displacement equations.  The displacement (uo)i of the point Xi represents a major part of the motion of all points within a rock body.  The motion (uo)i is called rigid-body translation because it does not describe the motion of particles or rock relative to each other but rather specifies that all particles follow the same path.   In this brief introduction we have only specified how individual points move during deformation.  We have not yet considered the relative motion of the points. 
The relative motion of the points describes the shape change of the shark’s tooth.  The change in shape of the rock is predicted by the displacement gradients which will be developed in the next lecture.