Stress at a point and notation

Stress at a point

Cut the body by a plane through point $P$. If a small area $\Delta A$ around $P$ carries a resultant force $\Delta \mathbf{F}$, the stress vector is

$$\mathbf{t} = \lim_{\Delta A \to 0} \frac{\Delta \mathbf{F}}{\Delta A}$$

It depends on the orientation of the plane. The state of stress at $P$ is fully defined by the stresses on three mutually perpendicular planes, i.e. by the stress tensor.

Notation and sign convention

$\sigma_{ij}$: first subscript $i$ = direction of the normal to the face; second subscript $j$ = direction of the stress component. Normal stresses are written $\sigma_x, \sigma_y, \sigma_z$, shear stresses $\tau_{xy}$, etc.

$$[\sigma] = \begin{bmatrix} \sigma_x & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_y & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_z \end{bmatrix}$$

• A face is positive if its outward normal points in a positive axis direction.
• A stress component is positive if it acts in the positive axis direction on a positive face, or in the negative direction on a negative face.
• So tension is positive and compression negative; positive shear points the same way as the axis on a positive face.

Proof of $\tau_{xy}=\tau_{yx}$

Take an element $dx \times dy \times dz$ in equilibrium under plane stress (no body moments).

• Right and left faces (area $dy\,dz$): shear $\tau_{xy}$ acts up on the right face and down on the left face, forming a couple $(\tau_{xy}\,dy\,dz)\,dx$ (anticlockwise).
• Top and bottom faces (area $dx\,dz$): shear $\tau_{yx}$ acts to the right on the top face and to the left on the bottom face, forming a couple $(\tau_{yx}\,dx\,dz)\,dy$ (clockwise).

Moments about the z-axis through the centre:

$$\begin{aligned} \sum M_z &= 0 \\ (\tau_{xy}\, dy\, dz)\, dx - (\tau_{yx}\, dx\, dz)\, dy &= 0 \\ \tau_{xy} &= \tau_{yx} \end{aligned}$$

In the same way $\tau_{yz}=\tau_{zy}$ and $\tau_{zx}=\tau_{xz}$. Hence the stress tensor is symmetric and has only six independent components, not nine.