We found a recent paper by the university of South Australia / New South Wales to be particularly interesting. Here’s a quick summary on this novel work and how we see its potential practical implications for geotechnical engineering applications.
Secondary compression — the continued settlement of soft clay under constant effective stress after primary consolidation is complete — is one of those phenomena that geotechnical engineers learn to account for, but rarely model with much sophistication. The standard approach treats the creep coefficient Cα as a material constant: plot void ratio against the logarithm of time, measure the slope, and apply it forward. It is simple, and for many practical purposes it is adequate. But long-duration test data tell a different story, and a paper by Karim and Lo (2020), published in Geotechnical Research, makes a careful case for why a more nuanced approach is both justified and achievable.
The problem with a constant Cα
The assumption underlying a constant Cα is that the void ratio–log(time) relationship is linear. This may be a reasonable approximation over the duration of a typical oedometer test — a few days at most — but it becomes progressively less defensible as the observation window extends. Long-duration creep tests, some running for months, consistently show that the rate of secondary compression decelerates with time: the soil creeps quickly at first and then progressively more slowly. A constant Cα, applied over decades, will therefore overestimate long-term settlement — sometimes substantially.
This non-linearity has been noted since at least the 1970s (Berre and Iversen, 1972; Leroueil et al., 1985; Mesri and Castro, 1987), and several researchers have proposed creep functions to capture it. Yin (1999) proposed a non-linear creep parameter for oedometer tests, and Karim et al. (2010) developed a tangential creep coefficient Cat that varies with the progression of creep, expressed as:
Cat = Camax · exp[−N(p̄₀ − p̄L)]
where p̄₀ is the creep-inclusive preconsolidation pressure (reflecting the current state of the soil including any quasi-preconsolidation developed by creep itself) and p̄L is the creep-exclusive preconsolidation pressure (essentially the yield stress in the absence of creep effects). The difference (p̄₀ − p̄L) measures how far creep has progressed: as creep proceeds under constant effective stress, p̄₀ increases relative to p̄L, the exponent grows, and Cat decreases. The soil is, in effect, becoming harder to creep further — a consequence of the quasi-preconsolidation concept introduced by Leonards and Altschaefl (1964) and formalised in the Bjerrum (1967) timeline framework.
What the new data show
The Karim et al. (2010) function was calibrated on soft clay from the Leneghans swamp embankment in New South Wales, and it performed well there. The question addressed by Karim and Lo (2020) is how well it generalises. The authors performed four new long-duration creep tests on reconstituted alumina tailings clays — red mud — from two sites in North Queensland, with markedly different index properties (liquid limits of 49% and 86%, plasticity indices of 40% and unknown). They supplemented these with published data from six long-duration creep tests on reconstituted Hong Kong marine deposits (Yin, 1999) spanning effective stresses from 50 to 800 kPa, including reloading tests.
When Cat is plotted against (p̄₀ − p̄L), the data from individual tests behave as expected — Cat decreases with progression of creep — but tests conducted at different stress levels form distinct groups that the original function cannot reconcile. At higher stresses, the (p̄₀ − p̄L) term becomes large but Cat does not drop as rapidly as the exponential form predicts, suggesting that the raw difference in preconsolidation pressures is not a stress-normalised quantity and therefore cannot adequately capture behaviour across a wide stress range.
A better parameter: p̄₀/p̄L
The resolution the authors propose is straightforward and intuitive: normalise (p̄₀ − p̄L) by p̄L, replacing the absolute difference with the ratio p̄₀/p̄L. The revised Cat function becomes:
Cat = Cam · exp[−A√(p̄₀/p̄L − 1)]
where Cam and A are two positive constants calibrated from long-duration creep tests. When p̄₀/p̄L = 1, the soil has just yielded and Cat is at its maximum value Cam. As creep proceeds and p̄₀/p̄L increases, Cat decays smoothly toward zero. Crucially, because p̄₀/p̄L is dimensionless and stress-normalised, the function collapses data from tests conducted over a wide range of effective stresses onto a single trend — something the original (p̄₀ − p̄L) form could not achieve.
The improvement is tangible. For the Hong Kong marine deposit data, the root-mean-square deviation of Cat around the best-fit curve falls from 0.012 using the original function to 0.0078 using the revised one — a 35% reduction in scatter. For the Leneghans swamp data replotted on the new basis, tests at 275, 350 and 929 kPa — a more than threefold stress range — now sit on a single coherent trend with essentially the same scatter as before (RMSD = 0.0090). The function generalises without degradation.
Why this matters
The practical implications are real. Settlement predictions for structures on soft clay are only as good as the constitutive model used to describe creep behaviour. A constant Cα, applied over a design life of 50 or 100 years, will in general be too pessimistic in the long term — creep decelerates, and a well-calibrated Cat function captures this. The function proposed by Karim and Lo (2020) requires long-duration oedometer tests (at least one week, preferably longer) conducted at two or more stress levels to calibrate Cam and A, alongside standard modified Cam Clay parameters to compute p̄₀ and p̄L. It is not tied to any specific elastic-viscoplastic model, which means it can in principle be adopted within most EVP frameworks already in use.
The deeper point is the one that makes the p̄₀/p̄L formulation physically satisfying: it treats creep deceleration not as a function of time, but as a function of the soil’s evolving state. Time-based formulations are inherently awkward because time is not a state variable — the same soil at the same state will behave the same way regardless of how long it took to get there. A state-based measure, expressed through the ratio of creep-inclusive to creep-exclusive preconsolidation pressure, avoids this difficulty and is consistent with how modern constitutive models represent soil behaviour more generally.
References
Berre, T. and Iversen, K. (1972) Oedometer tests with different specimen heights on a clay exhibiting large secondary compression. Géotechnique, 22(1), pp. 53–70.
Bjerrum, L. (1967) Engineering geology of Norwegian normally consolidated marine clays as related to settlements of buildings. Géotechnique, 17(2), pp. 83–118.
Karim, M.R., Gnanendran, C.T., Lo, S.C.R. and Mak, J. (2010) Predicting the long-term performance of a wide embankment on soft soil using an elastic-visco-plastic model. Canadian Geotechnical Journal, 47(2), pp. 244–257.
Karim, M.R. and Lo, S.C.R. (2020) Non-linearity of creep coefficient. Geotechnical Research, 7(2), pp. 90–95.
Leonards, G.A. and Altschaefl, A.G. (1964) Compressibility of clays. Journal of the American Society of Civil Engineers, 90(5), pp. 133–156.
Leroueil, S., Kabbaj, M., Tavenas, F. and Bouchard, R. (1985) Stress–strain–strain rate relation for the compressibility of sensitive natural clays. Géotechnique, 35(2), pp. 159–180.
Mesri, G. and Castro, A. (1987) Cα/Cc concept and K0 during secondary compression. Journal of Geotechnical Engineering, 113(3), pp. 130–247.
Yin, J.H. (1999) Non-linear creep of soils in oedometer tests. Géotechnique, 49(5), pp. 699–707.
