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Traction Vectors

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Introduction

The traction vector, \({\bf T}\), is simply the force vector on a cross-section divided by that cross-section's area.

\[ {\bf T} = { {\bf F} \over \text{Area} } \]
So \({\bf T}\) has units of stress, like MPa, but it is absolutely a vector, not a stress tensor. So all the usual rules for vectors apply to it. For example, dot products, cross products, and coordinate transforms can be applied.

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Normal and Shear Stresses

Normal and shear stresses are simply the components of the traction vector that are normal and parallel to the area's surface as shown in the figure. Using \({\bf n}\) for the unit normal vector to the surface, and \({\bf s}\) for the unit vector parallel to it, means that

\[ \sigma = {\bf T} \cdot {\bf n} \qquad \text{and} \qquad \tau = {\bf T} \cdot {\bf s} \]
It's very important to recognize that \(\sigma\) and \(\tau\) here are each scalar values, not full tensors. This is the natural result of the dot product operations involving \({\bf T}\), \({\bf n}\), and \({\bf s}\). (Dot products produce scalar results.)

The normal and shear stress values here are scalars rather than tensors because they are only two individual components of the full stress tensor.

Also, note that in 3-D, there are in fact an infinite number of \({\bf s}\) vectors parallel to the surface, each having a different component in-and-out of the page, so to speak. This is why it is common to specify one parallel to the page and a second perpendicular to it.

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Stress Tensors and Traction Vectors

The relationship between the traction vector and stress state at a point results directly from setting the sum of forces on an object equal to zero, i.e., imposing equilibrium.

\[ \sigma_{xx} \, A \, \cos \theta + \tau_{xy} \, A \, \sin \theta = T_x \, A \] \[ \tau_{xy} \, A \, \cos \theta + \sigma_{yy} \, A \, \sin \theta = T_y \, A \]
The area, \(A\), cancels out of both sides leaving

\[ \sigma_{xx} \, \cos \theta + \tau_{xy} \, \sin \theta = T_x \] \[ \tau_{xy} \, \cos \theta + \sigma_{yy} \, \sin \theta = T_y \]
but \(\cos \theta\) and \(\sin \theta\) are the components of the unit normal to the surface, \({\bf n} = (\cos \theta, \sin \theta)\), that \({\bf T}\) is acting on.

Replacing the \(\cos \theta\) and \(\sin \theta\) with \(n_x\) and \(n_y\) gives

\[ \sigma_{xx} \, n_x + \tau_{xy} \, n_y = T_x \] \[ \tau_{xy} \, n_x + \sigma_{yy} \, n_y = T_y \]
Both equations can be summarized as

\[ {\bf T} = \boldsymbol{\sigma} \cdot {\bf n} \]
or in tensor notation as

\[ T_i = \sigma_{ij} \, n_j \]
The above equations are very useful, compact, matrix and tensor notation representations of the equilibrium equations. The full equations, in 3-D, are

\[ \sigma_{xx} \, n_x + \tau_{xy} \, n_y + \tau_{xz} \, n_z = T_x \] \[ \tau_{yx} \, n_x + \sigma_{yy} \, n_y + \tau_{yz} \, n_z = T_y \] \[ \tau_{zx} \, n_x + \tau_{zy} \, n_y + \sigma_{zz} \, n_z = T_z \]
The tensor notation term, \(\sigma_{ij} \, n_j\), leads to nine separate stress components. For example, both \(\sigma_{xz}\) and \(\sigma_{zx}\) are present above, and both are always equal. This is in fact common in all equations involving stress and strain.

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Stress Transforms

This section introduces an aspect of coordinate transformations of stress tensors that is a subset of the general case, which comes later. It does so by combining different equations involving the traction vector.

Recall that the normal and shear stresses on a surface are related to the traction vector by

\[ \sigma = {\bf T} \cdot {\bf n} \qquad \text{and} \qquad \tau = {\bf T} \cdot {\bf s} \]
Recall that the normal and shear stresses here are just scalar quantities on the surface, not a full stress tensor.

But we also saw that the traction vector is related to the full stress tensor by

\[ {\bf T} = \boldsymbol{\sigma} \cdot {\bf n} \]
Substituting this equation for \({\bf T}\) into the above ones gives

\[ \sigma = {\bf n} \cdot \boldsymbol{\sigma} \cdot {\bf n} \qquad \text{and} \qquad \tau = {\bf s} \cdot \boldsymbol{\sigma} \cdot {\bf n} \]
In tensor notation, the equations are

\[ \sigma = \sigma_{ij} \, n_i \, n_j \qquad \text{and} \qquad \tau = \sigma_{ij} \, s_i \, n_j \]
These represent very useful relationships between the stress tensor in the global coordinate system and the normal and shear stress components at any other orientation.