Examples of yield criteria in plasticity theory (Von Mises, Tresca)

📚 Quick Study Guide

• 🧱 Yield criteria are used to determine when a solid material will begin to deform plastically under stress.
• 👨‍🏫 Von Mises criterion states that yielding occurs when the Von Mises stress ($\sigma_v$) reaches the yield strength ($\sigma_y$) of the material: $\sigma_v = \sigma_y$.
• 📐 The Von Mises stress is calculated as: $\sigma_v = \sqrt{\frac{1}{2}[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]}$ where $\sigma_1$, $\sigma_2$, and $\sigma_3$ are the principal stresses.
• 🔨 Tresca criterion, also known as the maximum shear stress criterion, states that yielding occurs when the maximum shear stress ($\tau_{max}$) reaches half of the yield strength ($\sigma_y$): $\tau_{max} = \frac{\sigma_y}{2}$.
• 🧮 The maximum shear stress is calculated as: $\tau_{max} = \frac{\sigma_{max} - \sigma_{min}}{2}$, where $\sigma_{max}$ and $\sigma_{min}$ are the maximum and minimum principal stresses, respectively.
• 📊 Both criteria are used extensively in engineering design to predict material failure under complex loading conditions.

🧪 Practice Quiz

1. Which criterion states that yielding occurs when the Von Mises stress reaches the yield strength of the material?

   1. Rankine Criterion
   2. Tresca Criterion
   3. Von Mises Criterion
   4. Maximum Principal Stress Criterion

2. What is the formula for Von Mises stress ($\sigma_v$) in terms of principal stresses ($\sigma_1, \sigma_2, \sigma_3$)?

   1. $\sigma_v = \sqrt{(\sigma_1 + \sigma_2 + \sigma_3)^2}$
   2. $\sigma_v = \frac{1}{2}[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]$
   3. $\sigma_v = \sqrt{\frac{1}{2}[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]}$
   4. $\sigma_v = |\sigma_1 + \sigma_2 + \sigma_3|$

3. According to the Tresca criterion, yielding occurs when the maximum shear stress reaches:

   1. The yield strength ($\sigma_y$)
   2. Half of the yield strength ($\frac{\sigma_y}{2}$)
   3. Twice the yield strength ($2\sigma_y$)
   4. Zero

4. What is another name for the Tresca criterion?

   1. Maximum Normal Stress Criterion
   2. Maximum Shear Stress Criterion
   3. Octahedral Shear Stress Criterion
   4. Distortion Energy Criterion

5. In the Tresca criterion, how is the maximum shear stress ($\tau_{max}$) calculated?

   1. $\tau_{max} = \frac{\sigma_{max} + \sigma_{min}}{2}$
   2. $\tau_{max} = \sigma_{max} - \sigma_{min}$
   3. $\tau_{max} = \frac{\sigma_{max} - \sigma_{min}}{2}$
   4. $\tau_{max} = \sigma_{max} + \sigma_{min}$

6. Which of the following is a key difference between the Von Mises and Tresca criteria?

   1. Von Mises considers only maximum shear stress, while Tresca considers distortion energy.
   2. Tresca considers only hydrostatic stress, while Von Mises considers deviatoric stress.
   3. Von Mises is independent of the intermediate principal stress, while Tresca considers it.
   4. Tresca is independent of the intermediate principal stress, while Von Mises considers it.

7. Which criterion is generally considered more accurate for ductile metals?

   1. Tresca Criterion
   2. Von Mises Criterion
   3. Maximum Principal Stress Criterion
   4. Maximum Normal Stress Criterion