Examples of stress intensity factors for different crack geometries

📚 Quick Study Guide

• 🔍 Stress Intensity Factor (K): A parameter that characterizes the stress field near the tip of a crack. It depends on the applied load, crack size, and geometry.
• 📏 Units: Typically expressed in $MPa \sqrt{m}$ or $ksi \sqrt{in}$.
• 📐 Mode I (Opening Mode): Tensile stress applied perpendicular to the crack plane. This is the most common mode.
  • 📝 Formula: $K_I = \sigma \sqrt{\pi a} F$, where $\sigma$ is the applied stress, $a$ is the crack length, and $F$ is a geometry factor.
• ✂️ Mode II (Sliding Mode): Shear stress applied parallel to the crack plane and perpendicular to the crack front.
  • 🔬 Formula: $K_{II} = \tau \sqrt{\pi a} F$, where $\tau$ is the applied shear stress, $a$ is the crack length, and $F$ is a geometry factor.
• 扭 Mode III (Tearing Mode): Shear stress applied parallel to both the crack plane and the crack front.
  • 🔩 Formula: $K_{III} = \tau \sqrt{\pi a} F$, where $\tau$ is the applied shear stress, $a$ is the crack length, and $F$ is a geometry factor.
• 💡 Geometry Factor (F): A dimensionless factor that accounts for the specific geometry of the cracked body and the loading conditions. It's often found in handbooks or determined through numerical analysis.
• 📚 Common Crack Geometries and their Geometry Factors:
  • 📈 Center-cracked Plate: $F$ depends on the ratio of crack length to plate width ($a/W$).
  • 📍 Edge-cracked Plate: $F$ also depends on $a/W$, but the specific values are different from the center-cracked plate.
  • 🧮 Circular Crack: $K = \frac{2}{\sqrt{\pi}} \sigma \sqrt{a}$

Practice Quiz

1. Which mode of crack opening involves tensile stress applied perpendicular to the crack plane?
   1. Mode I
   2. Mode II
   3. Mode III
   4. Mixed Mode
2. What are the typical units for the stress intensity factor?
   1. $N/m^2$
   2. $MPa \sqrt{m}$
   3. $m/s$
   4. $kg/m^3$
3. What does the geometry factor (F) account for in the stress intensity factor equation?
   1. The applied load only
   2. The crack size only
   3. The specific geometry of the cracked body and loading conditions
   4. The material properties only
4. In a center-cracked plate, what parameter does the geometry factor (F) primarily depend on?
   1. The applied stress
   2. The crack length (a)
   3. The ratio of crack length to plate width (a/W)
   4. The material's Young's modulus
5. Which of the following formulas represents the stress intensity factor ($K_I$) for Mode I loading, where $\sigma$ is the applied stress, $a$ is the crack length, and $F$ is the geometry factor?
   1. $K_I = \sigma a F$
   2. $K_I = \sigma \sqrt{\pi a} F$
   3. $K_I = \sigma \pi a F$
   4. $K_I = \sqrt{\sigma \pi a} F$
6. For a circular crack, what is the stress intensity factor (K) if the applied stress is $\sigma$ and the crack radius is $a$?
   1. $K = \sigma \sqrt{a}$
   2. $K = \frac{2}{\sqrt{\pi}} \sigma a$
   3. $K = \frac{2}{\sqrt{\pi}} \sigma \sqrt{a}$
   4. $K = \sigma a^2$
7. In Mode III crack opening (tearing mode), what type of stress is applied?
   1. Tensile stress
   2. Shear stress parallel to the crack plane and crack front
   3. Shear stress parallel to the crack plane and perpendicular to the crack front
   4. Compressive stress