One-dimensional consolidation — the coefficient of compressibility of soil (a_v) is defined as the ratio of:

Introduction / Context:
Compressibility parameters quantify how soils deform under changes in effective stress during consolidation. The coefficient of compressibility a_v is a fundamental slope parameter measured in oedometer (one-dimensional consolidation) tests and is used in settlement analysis and in computing other indices such as the coefficient of volume compressibility m_v and constrained modulus M.

Given Data / Assumptions:

• One-dimensional loading with lateral strain constrained (oedometer condition).
• Small strain increments over which linearization is reasonable.
• Effective stress changes are considered (drained consolidation).

Concept / Approach:

The coefficient of compressibility a_v is defined as vertical strain per unit increase in effective stress. In differential form: a_v = dε_v / dσʹ. Units are typically 1/stress (e.g., m^2/kN if expressed via thickness). Closely related, the coefficient of volume compressibility m_v is settlement per unit thickness per unit stress and is numerically equal to a_v in 1D consolidation where ε_v = settlement/thickness.

Step-by-Step Solution:

Verification / Alternative check:

Relate to constrained modulus M = 1 / a_v in 1D; cross-check with e-log σʹ curve to ensure consistency of compressibility across stress ranges.

Why Other Options Are Wrong:

Stress/strain is stiffness (modulus), not compressibility.

Stress/settlement and rate ratios are not standard definitions of a_v.

Void ratio change to stress change describes m_v via volume terms but is not the direct a_v definition as posed.

Common Pitfalls:

Using total stress instead of effective stress; mixing settlement with strain without normalizing by layer thickness; assuming linearity over large stress jumps.

Final Answer:

strain to stress