Structural Geology
Lecture 6
A Second Derivation of the Displacement Gradient
(another look at strain)

Three examples of deformation mapping were given in the previous lecture.  The deformation can be represented in two manners.  The displacement vector u was mapped in one case leading to a displacement gradient.  In the other case the new position of each particle was mapped based on the initial position of the particle leading to a deformation gradient. In this lecture we will show that in 2 and 3 dimensions deformation consists of rotations as well as stretches.

We start by taking another look at strain with some simple definitions such as a change in length of line per unit length of line. 
 
	e =  Dl/l

This is equivalent to a stretch. A formal definition of shear strain (g) is the change in angle (y) between two initially perpendicular lines (Fig. 6-1).

	g  =  tan y

(Fig. 6-1)

A second measure of shear strain is the tensor shear strain which is half the tangent of the change in angle between initially perpendicular lines.

	Tensor shear strain  =  g/2.

g is sometimes called the engineering shear strain.  Note here that shear strain is represented by line rotations.  This gives the first indication that the strain can be separated into a rotational and irrotational component.  We will deal in more detail with these concepts in the next lecture.

The concept of strain in one dimension uses l0 to indicate the initial length of a line and l1 the final length of a line.  We will start with the following definitions

	e  =  Dl/l0                  	(elongation)

	S  =  l1/l0  =  (1 + e)	(stretch)

	l  =  (l⁄/l0)2  =  (1 + e)2 	(quadratic elongation)

	e  =   dl/l0	(infinitesimal strain)

	e  =  Dl/l0           	 (small increment of strain)

                                       l1                                                                                                                                                                                             
	e  =  ∫  dl/l0  =  ln (l1/l0)  =  ln(1 +  e)  =  1/2lnl
	         l0                                     	
		(natural strain)

Now let's take another look at the displacement gradients by looking at how a rectangular element at P is distorted as shown in Figure 6-2.

Figure 6-2

The meaning of u1 and u2 is as shown above.  Stretch of PQ1 parallel to the 1 axis is:

Anticlockwise rotation of PQ1 is given as 

The angle through which PQ1 turns is:

Because Du1  <<  Dx1:  

For no distortion the following situation holds:

Figure 6-3