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


Tensor Notation

Suffix notation

Suffices are used to represent components of tensors and vectors. For example in the case of a vector x = (x1 x2 x3) we can then refer to its jth component as xj. We can also refer to x as the vector xj where we know that j can take the values 1, 2 and 3 ( j is then known as a free suffix).

It is important to note that tensors are defined with respect to a basis just like with vectors, and that the individual components of the tensor change when the basis is changed, while the magnitude and physical meaning stay the same. Note that there are different conventions for the order of the suffices. In this TLP we use the tensor component Tij to represent the effect on the i axis due to action on the j axis.

Einstein summation convention

Let us consider the equation: x.y = x1y1 + x2y2 + x3y3. This can be written as x.y = 3i=0 xiyi.

Using the Einstein summation convention, we can drop the sigma and just write this as x.y = xiyi, remembering to sum over all the indices. Another example of this is the equation y = (a.b)x, which can be written using the summation convention as yi = ajbjxi where j is summed over (known as a dummy suffix) and the value of i can be 1, 2 or 3 (i.e. is a free suffix). Note that in effect this represents 3 separate equations, one for each vector component.

If a suffix appears twice in a term it is a dummy suffix and is summed over, whereas free suffices appear once in every term.

A more complex example is: (|a|2 - c.a)x + |b|2y = zφ can be rewritten as (ajaj  clal)xi + bkbkyi = ziφ.

Second rank tensors have components in two directions. This leads to the components of the tensor A being written aij such that a tensor operating on a vector to give another vector y = Ax can be written yi = aijxj where we see that the suffix j is summed over. This also applies for tensor multiplication, for which C = AB becomes cij = aikbkj where k is summed over. Making use of this convention is a useful simplifying technique in proving tensor and vector properties.

Voigt Notation

As we have seen, many physical quantities are described by symmetric tensors. Voigt notation (also known as matrix notation) is an alternative way of representing and simplifying these tensors. An example using a symmetrical second rank tensor (e.g. stress) is shown below:

T11 T12 T13
T12 T22 T23
T13 T23 T33
T1 T6 T5
. T2 T4
. . T3

Tensor Notation 11 22 33 23,32 13,31 12,21
Voigt Notation 1 2 3 4 5 6

These substitutions allow us to represent a symmetric second rank tensor as a 6-component vector. Likewise a third rank tensor can be represented as a 3×6 matrix (keeping the first suffix e.g. T123 = T14), and a fourth rank tensor as a 6×6 matrix (doing the operation on the first two and then the last two suffices e.g. T1322 = T52). This is very useful as we can display every tensor up to 4th rank as a single two-dimensional matrix, simplifying the maths and making them easier to visualise. It is particularly useful for the equations of elasticity where σij = Cijklεkl can be converted to σi = Cijεj:

C11 C12 C13 C14 C15 C16
C21 C22 C23 C24 C25 C26
C31 C32 C33 C34 C35 C36
C41 C42 C43 C44 C45 C46
C51 C52 C53 C54 C55 C56
C61 C62 C63 C64 C65 C66

It should be noted that for convenience some scaling factors are often introduced when converting tensors into Voigt notation. For example, by convention the off-diagonal (shear) components of the strain tensor ε are converted such that in Voigt notation they are equal to the engineering shear strain:

ε11 ε12 ε13
ε21 ε22 ε23
ε31 ε32 ε33
ε1 ½γ12 ½γ13
½γ21 ε2 ½γ23
½γ31 ½γ32 ε3
ε1 ½ε6 ½ε5
. ε2 ½ε4
. . ε3
ε1 ε2 ε3 ½ε4 ½ε5 ½ε6

As such, care must be taken when looking up numerical values and converting between notations to check that consistent definitions are used.