Elastic constants relationship: if a material has modulus of elasticity E equal to twice its modulus of rigidity C (i.e., E = 2C), what is the corresponding bulk modulus K for that material?

Introduction / Context:Designers often convert between the three fundamental elastic constants for isotropic materials: Young’s modulus E, modulus of rigidity C (also written G), and bulk modulus K. This question checks your fluency in using the standard relations to move from one constant to another.

Given Data / Assumptions:

• E = 2C (i.e., E = 2G).
• Material is homogeneous, linear-elastic, and isotropic.
• Standard relationships among E, C, K, and Poisson’s ratio ν apply.

Concept / Approach:For isotropic materials, the following formulas hold:

• E = 2C * (1 + ν)
• E = 3K * (1 − 2ν)
• Alternatively, K = E / (3 * (1 − 2ν)) and C = E / (2 * (1 + ν))

Step-by-Step Solution:

Verification / Alternative check:Using the well-known identity E = 9KC / (3K + C). Set E = 2C and solve for K to confirm K = 2C/3.

Why Other Options Are Wrong:2C, 3C: These ignore the required division by 3 from the E–K–ν relationship.3C/2, C/2: Do not satisfy the isotropic consistency conditions when E = 2C.

Common Pitfalls:Forgetting that ν is implicated through E = 2C(1 + ν); if E = 2C, it forces ν = 0, not any arbitrary value. Always solve for ν first when constants are related in this way.

Final Answer:

2C/3