Strain Energy in Volumetric Compression For a body subjected to uniform hydrostatic pressure p producing volumetric strain, how does the strain energy due to volumetric strain vary with volume, pressure, and bulk modulus?

Introduction / Context:Strain energy is the elastic energy stored in a body due to deformation. Under uniform hydrostatic pressure, the energy associated with volumetric strain depends on pressure, bulk modulus, and the volume of the body.

Given Data / Assumptions:

• Hydrostatic pressure p.
• Bulk modulus K of the material.
• Body volume V.

Concept / Approach:For volumetric compression, the classic result is U_v = (p^2 * V) / (2 * K), assuming linear elasticity and small strains. This expression reveals the proportionalities directly.

Step-by-Step Solution:Start from U = 1/2 * stress * strain * volume for linear elastic response.For hydrostatic loading: volumetric strain = p / K.Thus U_v = 1/2 * p * (p / K) * V = (p^2 * V) / (2 * K).

Verification / Alternative check:Dimensional analysis confirms energy dimensions are consistent when combining p^2, V, and K in this form.

Why Other Options Are Wrong:

• Each individual statement a–c is true; the most complete option is “All of the above”.

Common Pitfalls:Mixing up bulk modulus in numerator rather than denominator or forgetting the square on pressure leads to incorrect trends.

Final Answer:All of the above