Long-Term Creep at Matched Secondary-Compression Slopes: Fractional Kelvin–Voigt and Finite-Band Becker Models

Secondary compression of soils shows a strain roughly linear in log time over several decades, with a slope—the coefficient of secondary compression C_α—that the Mesri law ties to the compression index C_c. Fractional creep elements are widely used for soil and rock creep; we ask how well they, and classical spectral laws, hold that slope within a stated tolerance. No single Scott-Blair springpot or fractional Kelvin–Voigt element yields an exactly constant C_α: the springpot slope grows by 10^α per decade and the Kelvin–Voigt slope decays as t^−α. A uniform retardation spectrum on a finite band [τ₁, τ₂]—the Becker law with an upper cutoff—gives the closed form C_α(t) = σ₀ A ln10 (e^−t/τ₂ − e^−t/τ₁), so the plateau height and tolerance-defined width follow exactly; holding C_α within 1% over N decades needs a band about N + 2.66 decades wide. For the same target slope and four-decade window under one amplitude rule (peak slope = C∗), a near-degenerate springpot and the Becker law need unbounded creep, while a Kelvin–Voigt element of order α ≈ 0.044 needs a bounded long-time compliance about six times that of the finite band. The finite-band element is thermodynamically admissible as a positive superposition of Voigt modes. The results support choosing a creep kernel by the window it sustains and the long-term settlement it implies.