Maximum shear stress

On the faces of the principal element on the left-hand side of Figure 2.4, there are no shear stresses. On a face inclined at any other angle, both normal and shear stresses will act. On faces of different orientation it is found that the shear stresses will locally reach a maximum for three particular directions; these are the maximum shear stress planes and are illustrated in Figure 2.4. They are inclined at 45◦ to the principal directions and the maximum shear stresses can be found from the Mohr circle of stress, Figure . Normal stresses also act on these maximum shear stress planes, but these have not been shown in the diagram.

The three maximum shear stresses for the element are
τ1 =(σ1 − σ2)/2 ; τ2 =(σ2 − σ3)/2 ; τ3 =(σ3 − σ1)/2
From the discussion above, it might be anticipated that yielding would be dependent on the shear stresses in an element and the current value of the flow stress; i.e. that a yielding condition might be expressed as
f (τ1, τ2, τ3) = σf