Swelling (recompression) line using Terzaghi’s e–log σ′ relation: If Cs denotes the swelling/expansion (recompression) index, which empirical form correctly relates void ratio e to the change in effective vertical stress σ′ from a reference state (e0 at σ0′)?

Introduction / Context:
Natural clays exhibit a near-linear relationship between void ratio e and the logarithm of effective stress σ′ over certain ranges. On unloading–reloading (swelling/recompression), the slope is the swelling (recompression) index Cs (often denoted Cr), smaller than the compression index Cc on the virgin line. Correct sign and form are essential for settlement back-analysis and preloading design.

Given Data / Assumptions:

• Reference state: e = e0 at σ′ = σ0′.
• Small-strain unloading/reloading region where the e–log σ′ relation is approximately linear.
• Base-10 logarithm is used (consistent with many soil mechanics texts).

Concept / Approach:
On the swelling/recompression line, as σ′ increases above σ0′, e decreases; as σ′ decreases below σ0′, e increases. A negative sign in front of Cs * log10(σ′/σ0′) captures this behavior: if σ′ > σ0′, the log ratio is positive and e reduces; if σ′ < σ0′, the log ratio is negative and e increases (swells). Using Cs rather than Cc reflects the smaller slope during recompression.

Step-by-Step Solution:

Verification / Alternative check:
Oedometer data plotted on e–log σ′ commonly shows a steep virgin slope Cc and a flatter recompression slope Cs; extrapolations with the above form fit measured points.

Why Other Options Are Wrong:

• Option B sign is wrong: it predicts e increasing with increased σ′.
• Option C is a rearrangement but does not present the functional form; the standard expression is Option A.
• Option D uses an undefined symbol e° and wrong sign.
• Option E uses a linear σ′ term instead of log; not Terzaghi’s empirical form.

Common Pitfalls:
Mixing natural vs base-10 logs; confusing Cs with Cc; applying the relation outside its valid stress range.

Final Answer:
e = e0 − Cs * log10(σ′/σ0′)