Structural Geology
Lecture 5
Geological Deformation
(a first look at ‘real’ strain)

Having introduced the nature of deformation and displacement equations in the previous lectures it is time to consider three of the more famous examples of geological deformation.  In the following examples the problems are specified in terms of both the displacement and deformation equations.

Problem #1  The first deformation to occur in the history of a sedimentary rock is overburden compaction.   This is represented by flattening in the vertical direction with no deformation in the horizontal directions.  The following equations represent overburden compaction as shown in Figure 5-1.

	Displacement equations:

		u1  =   0X1  +  0X2 +  0X3
		u2  =   0X1  +  0X2  +  0X3
		u3  =   0X1  +  0X2  -  0.5X3

	Deformation equations:

		x1  =  1X1 +  0X2  +  0X3
		x2  =  0X1  + 1X2   +  0X3
		x3  =  0X1  +  0X2  +  0.5X3

(Fig. 5-1)

Problem 2#: If the initial body were turned on its side this deformation could represent a tectonic compaction.  The Martinsburg shales near Harrisburg, Pennsylvania, have been isoclinally folded.  Beds which were once flat-lying are now standing on end.  Assuming that the shales were deformed with no internal strain we wish to represent this motion.  The following equations represent the rigid rotation of a block by 90° as shown in Figure 5-2.

	Displacement equations (A vector from the old point to its new location):

		u1  =   0X1 + 0X2 + 0X3
		u2  =   0X1  -  1X2  - 1X3
		u3  =   0X1  +  1X2  -  1X3

	Deformation equations (A map of the new point given the coordinates of its old position):

		x1  =  1X1 +  0X2  +  0X3
		x2  =  0X1  + 0X2   -  1X3
		x3  =  0X1  +  1X2  +  0X3

These equations are represented in the following figure:

 
(Fig. 5-2)

For an example in Problem #2, the initial point (1,1,1) is mapped to the new point (1, -1, 1).  The vector between these two points is u = (0, -2, 0).

Problem #3:  As a final review of deformation equations, we shall take a look at the behavior of a fault zone subject to simple shear.   In higher grade terrains these fault zones are simply called shear zones.  Simple shear is illustrated by the deformation of a unit cube in Fig. 5-3.

Displacement equations (A vector from the old point to its new location):

		u1  =   0X1  +  0X2  +  0X3
		u2  =   0X1  +  0X2  +  0.5X3
		u3  =   0X1  +  0X2  +  0X3

	Deformation equations (A map of the new point given the coordinates of its old position):

		x1  =  1X1 +  0X2  +  0X3
		x2  =  0X1  + 1X2   +  0.5X3
		x3  =  0X1  +  0X2  +  1X3

These equations are represented in the following figure:

Figure 5-3