Question

The units of fracture toughness (K\(_{\text{IC}}\)) are

• A. MPa m\(^2\)
• B. MPa m\(^{1/2}\)
• C. N/m\(^2\)
• D. J/mol

Hint:

Double check units to avoid simple slip-ups:
- Fracture toughness: \( \text{MPa}\cdot\text{m}^{1/2} \) (or \( \text{MPa}\sqrt{\text{m}} \)).
- Strain energy release rate (\( G_C \)): \( \text{J/m}^2 \) (or N/m).
- Yield strength / stress: MPa (or \( \text{N/mm}^2 \)).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question asks for the standard SI units utilized to measure plane-strain fracture toughness (\( K_{\text{IC}} \)) in linear elastic fracture mechanics.

Step 2: Key Formula or Approach:
The stress intensity factor \( K \) near a crack tip is mathematically defined by:
\[ K = Y \sigma \sqrt{\pi a} \]
where:
\( Y \) is a dimensionless geometric factor.
\( \sigma \) is the nominal applied stress.
\( a \) is the crack length.
Critical fracture toughness \( K_{\text{IC}} \) represents the threshold value of \( K \) at which a material undergoes unstable brittle crack propagation.

Step 3: Detailed Explanation:

• Dimensional Analysis of the Formula:
- The unit of stress \( \sigma \) in SI is Megapascals (MPa) or Newtons per square meter (\( \text{N/m}^2 \)).
- The unit of crack length \( a \) is meters (m).
- Thus, the term \( \sqrt{a} \) has the unit of square root of meters (\( \text{m}^{1/2} \)).
- Combining these terms:
\[ \text{Units of } K_{\text{IC}} = \text{Stress} \times \sqrt{\text{Length}} = \text{MPa} \cdot \sqrt{\text{m}} = \text{MPa} \cdot \text{m}^{1/2} \]
- In base SI units, this is equivalent to:
\[ \text{N/m}^2 \cdot \text{m}^{1/2} = \text{N} \cdot \text{m}^{-3/2} = \text{Pa} \cdot \text{m}^{1/2} \]

• Other Options:
- \( \text{MPa}\cdot\text{m}^2 \) is dimensionally incorrect.
- \( \text{N/m}^2 \) is the unit for stress or pressure.
- J/mol is the unit for chemical potential or thermodynamic molar energy.


Step 4: Final Answer:
The correct units for plane-strain fracture toughness are \( \text{MPa}\cdot\text{m}^{1/2} \).
Therefore, the correct choice is option (B).