The current Hardening Soil small-strain (HSsmall) model, developed by Benz (2007), is capable of capturing the behaviour of a wide range of soil types, including their small-strain response. However, under specific conditions, the model exhibits overshooting, which is the overestimation of the material’s elastic stiffness upon the closure of a small unloading-reloading (UL-RL) cycle interrupting monotonic loading, rather than recovering the stiffness corresponding to the onset of the cycle, as observed in real soil behaviour. Overshooting leads to underestimated deformations and consequently non-conservative results.
To address this issue, PLAXIS (part of Seequent, the Bentley Subsurface company) proposed two formulations: the Continuous Brick (CB) formulation, which replaces the small-strain component of the HSsmall model, and a Memory-Surface-Based (MSB) formulation, which extends it. Both have been implemented in the source code of the existing HSsmall constitutive model. However, their implementations had not yet been verified in the literature, nor had an assessment of overshooting, the motivation behind their development, been conducted.
This research aims to close that gap and to determine which of the two formulations provides the most suitable approach. Implementing the best-performing formulation into the HSsmall model results in a new more robust state-of-the-art model.
Accordingly, the following research question is posed:
‘To what extent do the Continuous Brick formulation, as a new formulation for small-strain stiffness, and the Memory-Surface-based formulation, as an extension to the existing small-strain stiffness formulation within the Hardening Soil small-strain model, reduce the overshooting observed in the current formulation?’
This question is addressed through a structured test plan consisting of two main components: (i) single stress point simulations to verify whether the formulations behave as expected at the most fundamental level, and (ii) a boundary value problem to evaluate their performance under more numerically demanding conditions representative of practical applications in the pre-failure range, where small strain stiffness strongly influences the magnitude of deformations.
It was found that both formulations reduce overshooting to a negligible level. However, their effectiveness decreases in simulations involving nested cycles, as both formulations exhibit the limitation of retaining the memory of only a single UL-RL cycle. Although both formulations perform well, the MSB formulation proves to be the most suitable approach: it is easier to interpret, appears more robust, retains the small-strain component of the original HSsmall model, and yields a response consistent with the HSsmall model under monotonic loading.
Extending the HSsmall model with the MSB formulation leads to more accurate deformation estimations in geotechnical problems especially within the pre-failure range, as deformations will no longer be underestimated, without introducing additional model parameters. Moreover, adopting this formulation will not have major consequences for the end user, since its behaviour is consistent with that of the HSsmall model, except that overshooting no longer occurs.