Index Parameters – Find Void Ratio from a_v and m_v Relationship If the coefficient of compressibility a_v = 6.75 × 10^-2 and the coefficient of volume change m_v = 3 × 10^-2 (in consistent units), determine the void ratio e of the soil sample.

Introduction / Context:
The small-strain compressibility of saturated soils in one-dimensional loading is commonly expressed using two related parameters: the coefficient of compressibility a_v and the coefficient of volume change m_v. Their relationship involves the void ratio e and allows back-calculation of e from laboratory test data when direct measurement is unavailable or needs checking.

Given Data / Assumptions:

• a_v = 6.75 × 10^-2 (units consistent with stress increment).
• m_v = 3 × 10^-2 (same unit system).
• Small strain, saturated conditions; standard one-dimensional framework.

Concept / Approach:

The definitions are: a_v = Δe / Δσ′ and m_v = (Δe / (1 + e)) / Δσ′. Eliminating Δe / Δσ′ gives the relation m_v = a_v / (1 + e). Hence (1 + e) = a_v / m_v and e = (a_v / m_v) − 1.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional consistency is maintained because a_v and m_v share the same stress denominator; the ratio is dimensionless and equals (1 + e) as derived from definitions.

Why Other Options Are Wrong:

1.10–1.20 underestimate 1 + e; 1.30 slightly overestimates; only 1.25 satisfies the exact ratio.

Common Pitfalls:

Reversing a_v and m_v in the relation; mixing units; attempting to use compression index Cc (a different parameter) in place of a_v.

Final Answer:

1.25