Representing stress as a tensor

To understand this page, you first need to understand tensors! Good sources are the books by J.F. Nye [1], G.E. Dieter [2], and D.R. Lovett [3] referred to in the section Going Further in this TLP. Many undergraduate university courses in physical science or engineering have a series of lectures on tensors, such as the course at Cambridge University Department of Materials Science and Metallurgy, the handout for which can be found here.

The stress tensor is a field tensor – it depends on factors external to the material. In order for a stress not to move the material, the stress tensor must be symmetric: σij = σji – it has mirror symmetry about the diagonal.

The general form is thus:

$$\left( {\matrix{ {{\sigma _{11}}} & {{\sigma _{12}}} & {{\sigma _{31}}} \cr {{\sigma _{12}}} & {{\sigma _{22}}} & {{\sigma _{23}}} \cr {{\sigma _{31}}} & {{\sigma _{23}}} & {{\sigma _{33}}} \cr } } \right)$$ or, in an alternative notation, $$\left( {\matrix{ {{\sigma _{xx}}} & {{\tau _{xy}}} & {{\tau _{zx}}} \cr {{\tau _{xy}}} & {{\sigma _{yy}}} & {{\tau _{yz}}} \cr {{\tau _{zx}}} & {{\tau _{yz}}} & {{\sigma _{zz}}} \cr } } \right)$$

The general stress tensor has six independent components and could require us to do a lot of calculations. To make things easier it can be rotated into the principal stress tensor by a suitable change of axes.

Principal stresses

The magnitudes of the components of the stress tensor depend on how we have defined the orthogonal x1, x2 and x3 axes.

For every stress state, we can rotate the axes, so that the only non-zero components of the stress tensor are the ones along the diagonal:

$$\left( {\matrix{ {{\sigma _1}} & 0 & 0 \cr 0 & {{\sigma _2}} & 0 \cr 0 & 0 & {{\sigma _3}} \cr } } \right)$$

that is, there are no shear stress components, only normal stress components.

This is an example of a principal stress tensor of all the tensors we could use to express the stress state that exists. The elements σ1, σ2, σ3 are the principal stresses. The positions of the axes now are the principal axes. While it may be that σ1 > σ2 > σ3, it only matters that the x1, x2 and x3 axes define the directions of the principal stresses.

The largest principal stress is bigger than any of the components found from any other orientation of the axes. Therefore, if we need to find the largest stress component that the body is under, we simply need to diagonalise the stress tensor.

Remember – we have not changed the stress state, and we have not moved or changed the material – we have simply rotated the axes we are using and are looking at the stress state seen with respect to these new axes.

Hydrostatic and deviatoric components

The stress tensor can be separated into two components. One component is a hydrostatic or dilatational stress that acts to change the volume of the material only; the other is the deviatoric stress that acts to change the shape only.

$$\left( {\matrix{ {{\sigma _{11}}} & {{\sigma _{12}}} & {{\sigma _{31}}} \cr {{\sigma _{12}}} & {{\sigma _{22}}} & {{\sigma _{23}}} \cr {{\sigma _{31}}} & {{\sigma _{23}}} & {{\sigma _{33}}} \cr } } \right) = \left( {\matrix{ {{\sigma _H}} & 0 & 0 \cr 0 & {{\sigma _H}} & 0 \cr 0 & 0 & {{\sigma _H}} \cr } } \right) + \left( {\matrix{ {{\sigma _{11}} - {\sigma _H}} & {{\sigma _{12}}} & {{\sigma _{31}}} \cr {{\sigma _{12}}} & {{\sigma _{22}} - {\sigma _H}} & {{\sigma _{23}}} \cr {{\sigma _{31}}} & {{\sigma _{23}}} & {{\sigma _{33}} - {\sigma _H}} \cr } } \right)$$

where the hydrostatic stress is given by \({\sigma _H}\) = \({1 \over 3}\)\(\left( {{\sigma _1} + {\sigma _2} + {\sigma _3}} \right)\).

In crystalline metals plastic deformation occurs by slip, a volume-conserving process that changes the shape of a material through the action of shear stresses. On this basis, it might therefore be expected that the yield stress of a crystalline metal does not depend on the magnitude of the hydrostatic stress; this is in fact exactly what is observed experimentally.

In amorphous metals, a very slight dependence of the yield stress on the hydrostatic stress is found experimentally.