Volumetric strain under equal normal stresses on all faces A solid cube is subjected to equal normal stresses on all six faces (hydrostatic pressure/tri-axial equal loading). The volumetric strain will be x times the linear strain measured along any of the principal axes. What is x?

Introduction / Context:
When a body is subjected to equal normal stresses in three mutually perpendicular directions, the deformation is isotropic and the total volumetric change relates simply to the linear strains along each axis. This is a fundamental concept in strength of materials and elasticity relevant to pressure vessels and soil mechanics (volumetric strain).

Given Data / Assumptions:

• Equal normal stress applied along x, y, z directions.
• Small strains; linear elasticity.
• Linear strain ε is the same in magnitude along each principal axis (for equal stress), ignoring Poisson coupling for the ratio asked (since we compare total to any one component).

Concept / Approach:

Volumetric strain ε_v is defined as the sum of linear strains in three orthogonal directions: ε_v = ε_x + ε_y + ε_z. If all three linear strains are equal (ε_x = ε_y = ε_z = ε), then ε_v = 3ε. Therefore, the volumetric strain is three times the linear strain measured along any direction.

Step-by-Step Solution:

Verification / Alternative check:

This result is independent of elastic constants E and ν because the ratio is purely geometric for equal strains along orthogonal axes.

Why Other Options Are Wrong:

• 1 or 2 underestimate the contribution from three directions.
• 4 and 5 overestimate because there are only three orthogonal components being summed.

Common Pitfalls:

• Confusing the algebraic sum definition of volumetric strain with a product or average.

Final Answer:

3.