Dissemination of IT for the Promotion of Materials Science (DoITPoMS)


Principal axes

As we have seen, a general second rank tensor has the form:

T11 T12 T13
T21 T22 T23
T31 T32 T33

However, in a particular basis, this takes a simpler form:

T11' 0 0
0 T22' 0
0 0 T33'

i.e. All off diagonal elements are zero. These basis vectors are known as the principal axes (or directions) and the non-zero tensor components as the principal values. Note that in general the principal axes for a given property will not necessarily coincide with the crystal axes.

Using principal axes simplifies the mathematics and highlights the symmetry of the situation. Considering once again the case of electrical conductivity, when working in an arbitrary basis the equations take the form:

j1 = σ11E1 + σ12E2 + σ13E3
j2 = σ21E1 + σ22E2 + σ23E3
j3 = σ31E1 + σ32E2 + σ33E3

In the principal basis they take the form:

j1 = σ11E1
j2 = σ22E2
j3 = σ33E3

i.e. The effect of an action along a principal axis is also directed along that axis (the conductivities along each principal axis are of course different from each other).

Finding the principal axes

As we have seen, in the principal basis the component equations become Tx = λx where λ is a constant of proportionality. This represents 3 different linear equations where λ has 3 possible values (the principal values).

T11x1 + T12x2 + T13x3 = λx1
T21x2 + T22x2 + T23x3 = λx2
T31x3 + T32x2 + T33x3 = λx3

There is a useful solution for this when |T − λI| =  0, i.e. when:

T11  λ T12 T13
T21 T22  λ T23
T31 T32 T33  λ
 = 0

This gives a cubic equation in λ called the secular equation. To find the principal values we must solve this equation for λ. For each of the three solution for λ we find the vector x that solves the equation above. Each of theses solutions for x is a vector parallel to one of the principal axes. This vector can be of any length so long as it points along the principal axis, so generally we scale the vector so that it is of unit length, giving us an orthonormal basis.

It is worth noting that the principal values are called the eigenvalues of the matrix representing T, and the unit vectors along the principal axes are its eigenvectors. The general operation of finding these is not only useful when simplifying tensors, but is used throughout physics and chemistry for example in studying modes of vibration, and calculating energies in quantum mechanics. The activity below shows you how to find the secular equation and principal values of a symmetric second rank tensor.

If we are working with a tensor where the one of the principal values is given, i.e. a tensor of the form:

T = 
T11 T12 0
T21 T22 0
0 0 T33

Then we can use the Mohr's circle construct to geometrically find the two unknown principal values. This is demonstrated further in the Theory of Metal Forming TLP. Transforming our tensor into the principal basis Using what we know about transformation matrices, i.e. that rij = xi'.xj, we can see that the transformation matrix to rotate from the old into the principal basis is simply the matrix of normalised eigenvectors (e1, e2 and e3).

R = 
| | |
e1 e2 e3
| | |

Below are two more flash programs to show you another example of finding the principal values and principal axes.