Structural Geology
Lecture 9
Properties of Tensors
(Tensor Analysis)

Einstein summation convention -- Here the Einstein summation convention applies and the components s ij relate two vectors fi and lj in a linear manner.   

The components sij form a 3 x 3 matrix: 

In the study of vector analysis we have introduced three sorts of quantities: the scalar; the vector; and then a nine component quantity which relates two three-component vectors.  These three quantities are all called tensors where:

Tensor of zero rank  =  Scalar -- This is a single number unrelated to any axes of reference.
Tensor of first rank  =  Vector  --  This is specified by three numbers or components, each of which is associated with one of the axes of reference.
Tensor of second rank  =  (e.g. stress)  --  This is specificed by nine numbers, or components, each of which is associated with a pair of axes (taken in a particular order).

Recall that the 2 dimensional Mohr's diagram described stresses associated with a pair of axes.  In this sense the Mohr diagram is a visual representation of one-third of the second rank stress tensor.  A tensor has the unique property that it can be transformed without losing its value.  By this we mean that a tensor has conponents representing a physical quantity which retains its identity regardless of which coordinate system (e.g. rectangular, cylindrical, etc.) is used or how the axes of that coordinate system may be rotated.   The physical quantity is the same regardless of the direction from which it is viewed.

Tensor transformation --  The physical quantity, such as thermal conductivity, kij, is the same regardless of the set of axis chosen.   This is the chief characteristic of a tensor.  Transformation from one coordinate system to another depends on the direction cosine relating the axes of each of the coordinate systems. 

aij are the components of the transformation matrix where a12 is the cosine of the angle between x1 of the old coordinate system and x2' of the new coordinate system.    The complete transformation matrix is

	(9-1)

The first sugscript in the a's refers to the 'new' axes and the second to the 'old'.  There are only three independent direction cosines of the nine given in the transformation matrix.  The coordinates of a point (x1,x2,x3) can also be transformed to the coordinates (x1',x2',x3') using direction cosines in the same manner (Fig. 9-1)

x1'   =   a11x1    +      a12x2    +    a13x3 
             
x2'   =   a21x1    +      a22x2    +    a23x3 

x3'   =   a31x1    +      a32x2    +    a33x3.	(9-2)

Figure 9-1

Likewise the equations for the transformation of a vector from one coordinate system to another resemble the equations for transforming a point.

	(9-3)

where p'i are the components of the vector in the new coordinate system and pj are the components of the vector in the old coordinate system.  Here the free suffix is i and the dummy suffix, j, is together in the right-hand term.  Written in its full form equation 3-2 becomes

	(9-4)

We can reverse the transformation process to find the old vector in terms of the new vector

	(9-5)

Here the dummy suffix, j, is separated by the free suffix, i.

Relations between direction cosines  -  Since each row of the array (3-1) represents three direction cosines of a straight line with respect to a coordinate system, x1, x2, x3, the following equations hold true

	(9-6)

In a condensed version equation 3-5 is written

	(9-7)

Since eachpair of rows of the array 3-1 represents the direction cosines of two lines at right angles, we have a situation comparable to a scalar product of two vectors at right angles.

	(9-8)

We can combine the orthogonality relations into one equation

	(9-9)

where the symbol, dij, is called the Kronecker delta.

Second rank tensor -- In general, if a property T relates tow vectors P = [p1,p2,p3] and     Q = [q1,q2,q3] in such a way that

	(9-10)

where the components, T11, T12, .....are constants, then T11, T12, .....are said to form a second-rand tensor.

	(9-11)

The values of the coefficients T11, T12, .....depend on the orientation of the coordinate axes x1, x2, x3.  Now suppose we choose a new set of coordinate axes x1', x2', x3' related to the old axes by the direction cosines, aij.  If so, the vectors P and Q have new components pi' and qi'.  Next we find P' in terms of Q' the series of equations shown below (The arrow means "in terms of").

	(9-12)

The following equations permit the transformation

	(9-13)

	(9-14)

	(9-15)

Substituting

	(9-16)

or

	(9-17)

So

	(9-18)

This is the Transformation Law for a tensor of second rank.  Written out in full form each of its nine equations are rather long.  The general equation is

	(9-19)

Just as the components pi and qi transform to pi' and qi', when the axes are changed, the nine coefficients Tij transform to the nine coefficients T'ij.  In this example i and j are free suffixes and k and l are dummy suffixes.   A second rank tensor must transform.

At this point it is important to appreciate the characteristics of a stress vector (f) which is nothing more that the stress tensor (sij) expressed as a vector acting on a surface whose normal is the vector l.

The principal axes of a tensor - The equation for the surface of a quadric (i.e., a second degree surface such as an ellipsoid or hyperboloid) is written

	(9-20)

where Sij are coefficients.  Performing the summation with respect to i and j equation is written in full by assuming the symmetric condition which is that        Sij = Sji.

	
	(9-21)

The coefficients of equation 3-17 transform like the components of a symmetrical 2nd rank tensor.

	(9-22)

The theory of the transformation of a symmetrical 2nd rank tensor is thus identical with the theory of th transformation of a quadric.  Thus, the surface represented by equation 3-17 is called the representation quadric of a tensor, Sij.

An important property of a quadric is the possession of principal axes.  These are three directions at right angles such that, when the general quadric is referred to them as axes, its equation takes the simpler form

	(9-23)

When we refer thje general tensor to principal axes, we have reduced the tensor components to the matrix

	(9-24)

where S1, S2, S3  are called the principal components of the tensor [Sij].  The equation for a standard ellipsoid is written

	(9-25)

In this equation the semi-axes of the representation quadric are the lengths .

We now define a property of a principal axis as the radius vector which parallels the normal to the quadric surface.  The radius of the quadric is specified by Sijxj. The normal to the surface of the quadric is specified by the vector, xi.  The condition that the radius vector and the normal are parallel is satisfied provided that the corresponding components are proportional.  This condition is satisfied by the equation

	(9-26)

where l is a constant.  Equation 3-23 is three homogeneous linear equations in the variables xi.  

	(9-27)

The trivial solution to these equations is that xi = 0.  If so, these equations would be worthless.  For a non trivial solution the determinant of the components of these three equations must be zero.

	(9-28)

or

	(9-29)

This cubic equation is called the secular equation.  The three roots, l', l'', and l''', give the three possible values of l that ensure that equations 3-24 have non-zero solution.  Those values of l are also called eigenvalues or characteristic values of the matrix S.  Each of the roots defines a direction in which the radius vector of the quadric is parallel to the normal, that is, the direction of one of the principal axes.  These directions are called eigenvectors or characteristic vectors of S.

The direction of the principal axes are found by substituting, say l', into equation 3-24.  These three equations may be solved for the ratios x1': x2': x3'.  Knowing the ratios we can solve for the direction cosines of the unit vector by

	(9-30)