Question

For a tissue with Young's modulus 4 kPa and shear modulus 1.5 kPa, what is the value of the Poisson's ratio?

• A. \(\frac{1}{4}\)
• B. \(\frac{1}{5}\)
• C. \(\frac{1}{2}\)
• D. \(\frac{1}{3}\)

Hint:

Memorize the key relationships between elastic moduli for isotropic materials: \(E = 2G(1 + \nu)\) and \(E = 3K(1 - 2\nu)\), where K is the bulk modulus.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem provides the Young's modulus (\(E\)) and the shear modulus (\(G\)) for a material (tissue) and asks to calculate its Poisson's ratio (\(\nu\)).
Step 2: Key Formula or Approach:
For an isotropic elastic material, the relationship between Young's modulus (\(E\)), shear modulus (\(G\)), and Poisson's ratio (\(\nu\)) is given by the formula: \[ E = 2G(1 + \nu) \] Step 3: Detailed Explanation:
We are given the following values:

• Young's modulus, \(E = 4\) kPa
• Shear modulus, \(G = 1.5\) kPa

We substitute these values into the formula: \[ 4 = 2 \times 1.5 \times (1 + \nu) \] \[ 4 = 3 \times (1 + \nu) \] Now, we solve for \(\nu\): \[ \frac{4}{3} = 1 + \nu \] \[ \nu = \frac{4}{3} - 1 \] \[ \nu = \frac{4 - 3}{3} \] \[ \nu = \frac{1}{3} \] Step 4: Final Answer:
The value of the Poisson's ratio is \(\frac{1}{3}\).