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Derivation of yield ellipse aspect ratio

For plane stress, let the principal stresses be \({\sigma _1}\) and \({\sigma _2}\), with \({\sigma _3} = 0\).

The yield surfaces for the Tresca yield criterion and the von Mises yield criterion are shown below.

The Tresca yield surface is an irregular hexagon and the von Mises yield surface is an ellipse. The ratio of the length of the major and minor axes of this ellipse is \(\sqrt 3 {\rm{ :1}}\).

In the quadrant where \({\sigma _1} > 0\) and \({\sigma _2} > 0\), the Tresca yield surface is a square.

To see this, first suppose \({\sigma _1} > {\sigma _2}\) for example. Since \({\sigma _3} < {\sigma _2}\), yield occurs on the Tresca criterion when

\[{\sigma _1} - {\sigma _3} = Y\]

i.e., for \({\sigma _1} = Y\) because \({\sigma _3} = 0\). When \({\sigma _1} = {\sigma _2}\), yield occurs at \({\sigma _1} = {\sigma _2} = Y\). Similarly, for \({\sigma _1} < {\sigma _2}\) in this quadrant, yield occurs when \({\sigma _2} = Y\).

The shape of the Tresca yield surface in the quadrant where \({\sigma _1} > 0\) and \({\sigma _2} < 0\) is a straight line because the third principal stress \({\sigma _3}\) will be the intermediate principal stress. Hence in this quadrant the yield criterion becomes

\[\left| {{\sigma _1} - {\sigma _2}} \right| = Y\]

whence the straight line linking (-1,0) to (0,1) in the diagram. The shape of the Tresca yield surface in the remaining two quadrants follows similarly.

For plane stress the von Mises yield criterion becomes

$${\left( {{\sigma _1} - {\sigma _2}} \right)^2} + {\sigma _2}^2 + {\sigma _1}^2 = 2{Y^2}$$

which for Y = 1 becomes

$$2{\sigma _1}^2 + 2{\sigma _2}^2 - 2{\sigma _1}{\sigma _2} = 2$$

i.e.,

$${\sigma _1}^2 + {\sigma _2}^2 - {\sigma _1}{\sigma _2} = 1$$

Thus the yield surface for plane stress passes through (1,0), (1,1) (0,1) (-1,0) (-1,-1), and (0,-1). It also passes through the points

$$\left( { - \frac{1}{{\sqrt 3 }},\frac{1}{{\sqrt 3 }}} \right){\rm{ and }}\left( {\frac{1}{{\sqrt 3 }}, - \frac{1}{{\sqrt 3 }}} \right)$$

as can be seen by direct substitution in the yield condition for \({\sigma _1} = -{\sigma _2}\).

The directions

$$\left[ {{\rm{1,1}}} \right]{\rm{ and }}\left[ {\frac{1}{{\sqrt 3 }}, - \frac{1}{{\sqrt 3 }}} \right]$$

are orthogonal and their magnitudes define the length of the major and minor axes of this ellipse.

Hence, the ratio of the length of the major and minor axes of this ellipse is \(\sqrt 3 {\rm{ :1}}\).

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