Question

The figure shows two yield loci for an isotropic material in the \((\sigma_I,\sigma_{II})\) plane. Here \(\sigma_I,\sigma_{II}\) are principal stresses and \(\sigma_Y\) is the uniaxial tensile yield stress. Which statements are correct? 

• A. Criterion \(P\) represents the von Mises criterion
• B. Criterion \(Q\) represents the Tresca criterion
• C. Criterion \(P\) represents the Tresca criterion
• D. Criterion \(Q\) represents the von Mises criterion

Hint:

Remember: \(\tau_y\) at pure shear is \(\sigma_Y/2\) (Tresca) vs. \(\sigma_Y/\sqrt{3}\) (von Mises). Since \(1/2 < 1/\sqrt{3}\), the Tresca locus is inside the von Mises ellipse.

Answer 1

The Correct Option is A, B

Step 1: Known shapes of yield loci in the \((\sigma_1,\sigma_2)\) plane.
For an isotropic metal under plane stress: 

• von Mises (J$_2$) criterion: \( \sigma_\text{eq}^2=\sigma_1^2-\sigma_1\sigma_2+\sigma_2^2=\sigma_Y^2 \). This is an ellipse (smooth curve) in the \((\sigma_1,\sigma_2)\) plane passing through the uniaxial points \((\sigma_Y,0)\) and \((0,\sigma_Y)\).
• Tresca (maximum shear) criterion: \( \max(|\sigma_1-\sigma_2|,|\sigma_2|,|\sigma_1|)=\sigma_Y \). Its locus is a regular hexagon with corners on the axes at \(\pm\sigma_Y\) and on the lines \(\sigma_1=-\sigma_2\) at \(\pm \tfrac{\sigma_Y}{2}\) in pure shear.

Step 2: Size comparison (which curve is inside/outside).
In pure shear (\(\sigma_1=\tau,\,\sigma_2=-\tau\)): Tresca yields at \(\tau=\sigma_Y/2=0.5\,\sigma_Y\); von Mises yields at \(\tau=\sigma_Y/\sqrt{3}\approx0.577\,\sigma_Y\).
Hence Tresca predicts yield earlier (more conservative) \(\Rightarrow\) its hexagon is the smaller curve, lying inside the von Mises ellipse. 

Step 3: Identify \(P\) and \(Q\) on the given plot.
In the figure, the solid outer (larger, smooth) locus is labeled \(\,P\), and the inner dashed locus is labeled \(\,Q\). Therefore: \[ P \;\text{(outer, smooth ellipse)} \Rightarrow \text{von Mises},\qquad Q \;\text{(inner, hexagon-like)} \Rightarrow \text{Tresca}. \] \[ \boxed{\text{Correct statements: (A) and (B).}} \]