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