Structural Geology
Lecture 4
Displacement and Deformation Equations
(kinematic analyses)

Kinematics, as you learned when taking elementary physics, is a mathematical description of the motion of objects.  In the case of structural geology kinematics is the description of the path that rocks took during deformation.  It is also the mathematical description of the relative position of two infinitesimal points during the deformation of rocks.  Two points can change by translating together, rotating around each other, or changing in distance relative to one another.  We shall call such a mathematical description deformation mapping.

(Fig. 4-1:  Until this figure is redrawn and xi = X’i)

The deformation gradient

If during rigid-body translation the particles of rock move relative to each other we must devise other equations to account for their relative motion.  To the undeformed state we can attach a line segment dX whose components are dXi.  The study of deformation is concerned with the change in orientation and length of dX as the point at X is moved by deformation to the point at x.  We say that the vector dXi is both stretched and rotated to become the new vector dxi.  To account for the relative motion of particles within the rock, we consider how the motion ui of any vector dX differs from the motion (uo)i of the vector X  (Fig. 4-1).  If
	xi  =  f(Xj)                                                                         
then
	xi  +  Dxi  =  f( Xj  +  DXj )                                                

By Taylor's expansion 

so
	Dxi  =   ( ¶xi/¶Xj) DXj.                                                           

¶xi/¶Xj  are coefficients called the deformation gradient and are a function of the location within the rock (Xi).  Referring back to the deformation of the shark’s tooth as discussed in lecture #3:

	
The displacement of the vector ui is the summation of (uo)i plus Eijdx where Eij (a displacement gradient) is a function of position within the rock.
   
	Eij   =   ¶ui/¶xj                                                                  

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 deformation gradients (Fig. 4-4):
	
	x1  =  5  +  X1  -  X2
	
	x2  =        2X1  +  2X2
	
	x3  =  X3 

In geology the displacement equations prove to be more useful than the deformation equations.  However, for the purpose of an introduction to strain analysis a brief introduction of the deformation equations is instructive.  Now it is time to consider the characteristics of the displacement equations.

Strain is represented in the same manner as the deformation of shapes which was presented in the previous lecture.  Lets examine the problem of representing strain in three dimensions using the coordinate system x1,x2,x3.  We define a point P(x1,x2,x3) which has been displaced to the point P'(x1,x2,x3) (Fig. 4-2).  The displacement vector between P and P' is uP where

	uP  =  ux1' i  +  ux2' j  +  ux3' k. 

(Fig. 4-2)

As is the case for one dimension strain in rocks is mapped by following two points, so we need an additional point Q which is displaced to Q'.  Its displacement vector is then

	uQ  =  ux1" i  +  ux2" j  +  ux3" k.

Now let A be the vector from P to Q and A' be the vector from P' to Q' (Fig. 4-3).  Here strain is the difference between A and A' which is represented as

	dA  =  A - A'.                                                                       

(Fig. 4-3)

dA/dx is the rate of change of displacement with position within the rock.  By vector analysis we have
	
		uP  +  A'  =  A  +  uQ 
                                                          
and
	
	dA  =  uQ  -  uP                                                                    

A function u is said to be analytic at a point P if it can be represented by a power series of powers of P - a with a radius of convergence R > 0.  Every analytic function can be represented by a power series.  Because u is an analytic function we can express uQ in terms of uP by employing a Taylor series expansion of uQ about P.  Remember that a Taylor expansion takes the following form:

	f(z)  =  f(a)  +  f'(a).(z - a)  +  (1/2!)f''(a).(z - a)2  + ........      

This is a series centered at a.  So the expansion of uQ about uP is
	                      
		uQ    =     uP    +     (∂u/∂X1)PDX1     +   (∂u/∂X2)PDX2   + 
   
							(∂u/∂X3)PDX3    +    ........ 

where  DX1  =   A1  (i.e. the components of A).  Here higher order terms may be neglected because of there small value.

 Using the summation of indices as discussed in the appendix

	uQ    =     uP    +     (∂u/∂Xj)PAj         j  =  1,2,3

Now

	dA   =     (∂u/∂Xj)PAj 

By convection

	(dA)i  =  dAi 
  
and
   
	(∂u/∂Xj)i   =   (∂ui/∂Xj).

So

	dAi  =  (∂ui/∂Xj).Aj.

∂ui/∂Xj  is called the displacement gradient.  These scalar quantities are the components of the displacement equations which take the following form:

	

	

As a quick aside the derivation of the displacement gradient can also be illustrated using pages 94 and 95 of Nye (1957).  In the three dimensional case displacement ui, are linearly related to the independent variables, in this case vectors dxj, by simultaneous equations
	
	u1   =   (u0)1  +   E11dX1  +  E12dX2  +   E13dX3
	
	u2   =   (u0)2  +   E21dX1   +  E22dX2  +   E23dX3
	
	u3   =   (u0)3  +   E31dX1   +  E32dX2  +   E33dX3  

or
	ui  =  (uo)i  +  EijdXj. 
                                                       
Eij are nine numbers in a  3 x 3 matrix that describe all motion within a rock that is not rigid-body translation.  The terms Eij are called displacement gradients.  As pointed out in the previous lecture a term with two suffixes relates two vectors.  In this example the vectors are the displacement u and the position vector dXi.  The Einstein summation convention as introduced in Lecture 2 applies to these equations for displacement.

Displacement and deformation gradients can be distinguished by the nature of the information that they impart.  The displacement equations indicate the direction of motion of particles during the deformation.  The vector u (u1, u2, u3) points to the final position of the particle from the initial position of the particle given the initial coordinate system (X1, X2, X3).  The displacement gradients (¶ui/¶xj) are a description of how the movement of the initial point varies with position in the rock.  In contrast, the deformation equations map the final position of a particle of rock (x1, x2, x3 ) in terms of the initial coordinate system (X1, X2, X3 ). 

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.   In terms of the displacement gradients the stretch is denoted as

	eij  =  1/2 (Eij  +  Eij)                                                         

and the rigid-body rotation is denoted as
 
	wij  =  1/2 (Eij  -  Eij).