Other empirical approaches

There are many other empirical correlations and approaches. Plasticity-based approaches are presented by Prevost (1985) and Elgamal et al. (2003). There are also approaches based on energy or damage parameters (Berril and Davis, 1985; Ahn and Park, 2013;

Azeiteiro et al., 2017). They can consider both shear stress and shear strain and hence reproduce results for irregular loading. However, there are numerous mathematical models most of which are not applicable for the intended use within this thesis. They are not compatible for integration into the developed explicit method, where simple contour plots are more advantageous.

4 State of the art modelling methods

The excess pore pressure accumulation and the general cyclic soil response are not only dependent on the factors presented in Chapter 3, but also on the interaction of genera-tion and dissipagenera-tion rates as well as spatial influences. This is for instance the layering or the interaction of structure and soil. These site-specific boundary conditions can only be included by using some kind of approximation of the field condition, which mostly involve numerical models. There are different approaches how the numerical model performs the calculation. For example, all cycles can be calculated individually, or the element re-sponse is estimated based on a combination of numerical calculation and cyclic laboratory test results. Additional effects result from the constitutive law or from calibration and regression effects.

4.1 Implicit numerical methods

Many engineering problems can be solved by investigating the monotonic foundation response with numerical simulations. This is often done with simple constitutive models such as the Mohr-Coulomb or the Drucker-Prager material law in two as well as in three dimensions. However, in some cases more advanced numerical tools are needed in order to investigate the stress-strain response for the accurate prediction of the soil response. The generation and dissipation of excess pore pressure, the redistribution of stresses as well as the development of plastic strains is of significant importance. Some problems require simple differentiation between monotonic and un- and reloading moduli which lead, for instance, to the use of the popular Hardening soil (small) model (Benz, 2007; Bentley Systems, 2022).

Sandy soils are an assembly of different particles with intrinsic parameters such as the shape, the particle bulk modulus, the grain size distribution and the particle density as well as state parameters such as void ratio. The general stress-strain relationship can already be described very well with advanced models for monotonic loading, but this goal has not yet been fully achieved for cyclic loading. Cyclic soil behaviour is extremely complex and can often only be partially approximated. First implicit models with cyclic characteristics have been created for example by Mrˇoz et al. (1978), Dafalias and Popov (1976), Ohno and Wang (1993). Similar approaches were already done by for instance Prevost (1977), Prevost et al. (1980) and Andersen et al. (1978) with a set of yield surfaces.

Prevost et al. (1980) were able to very well back-calculate load- as well as displacement-controlled triaxial and direct simple shear tests over a representative number of cycles.

However, these methods reach their limits not only with respect to the computational effort but also with respect to the possible accuracy when relatively large numbers of

load cycles (N > approx. 20) are to be considered (Safinus et al., 2011; Niemunis et al., 2005). With numerical simulation models, which use highly developed material laws – such as hypoplasticity with intergranular strain (Niemunis and Herle, 1998) or SANISAND (Dafalias and Manzari, 2004) – the system behaviour can be calculated in the best possible way. In sophisticated models, the overall soil response depends on pressure, load history and void ratio from which a correct contractive and dilative behaviour is simulated for different cyclic boundary problems. In general, there are many different soil models, of which the most common advanced approaches will be briefly presented.

Hypoplasticity and Hardening soil small model

For a long time the hypoplasticity model with intergranular strain (Kolymbas, 1988; Von Wolffersdorff, 1996; Gudehus, 1996; Niemunis and Herle, 1998) was the first choice when it came to sophisticated soil models for both monotonic and cyclic problems. In particu-lar, because the extension of Niemunis and Herle (1998) takes the realistic effects of soil behaviour under unloading and reloading into account. The model does not distinguish between elastic and plastic parts as there is no plastic yield surface. It requires 13 pa-rameters and is often compared with the Hardening Soil small model (HSsmall) (Marcher et al., 2000; Sheil and McCabe, 2016; Benz, 2007). The basic hypoplastic material model requires eight parameters and the extension for the intergranular strain concept five ad-ditional parameters. The HSsmall model is an elasto-plastic, stress-dependent, non-linear model with small strain consideration and isotropic hardening using 11 parameters. How-ever, the main advantage of hypoplasticity over the HSsmall is that it is state-dependent and soil densification is taken into account. In general, the stress level, the soil density, dilatancy, contractance and peak friction angle are considered with one single equation without a potential function for plastic or elastic deformation. Within the HSsmall model every accumulated displacement relates to stiffness degradation and not to an altered void ratio. This is the reason why the hypoplasticity was extensively used to investigate the soil response under cyclic loading (Taşan, 2011; Grabe et al., 2004). The shortcoming of the hypoplasticity model is that dilatancy only depends on stresses and therefore volu-metric strain and excess pore pressure can only be generated by stress changes and not by shearing (Niemunis and Herle, 1998).

The hypoplasticity model was frequently used in order to back-calculate small- or large-scale model tests. Taşan et al. (2010) and Grabe et al. (2005) used a two-phase model with a hypoplasticity model with intergranular strain for fully saturated soils to investigate the excess pore pressure within multiple cycles and were able to identify several influencing parameters. Taşan (2011) (Taşan, 2017; Taşan et al., 2010) identified influences of the number of cycles, the loading type, the relative density, the loading frequency, the soil permeability, and the pile diameter. Anyway, no method for practical estimation of excess pore pressure for site-specific conditions is given. Due to the soil-specific accumulation be-haviour and the soil-structure interaction, the results can only be transferred to boundary conditions of practical projects to a limited extent. Similar investigations were performed by Cuéllar (2011); Cuéllar et al. (2012, 2014).

4.1 Implicit numerical methods

Octahedral effective stress p [kPa]

Deviatoric stress q [kPa]

M^b M

M^d

a h

Void ratio e [1]

CSL

y= e-ec

ec

e

p

Octahedral effective stress p [kPa]

(a) (b)

Figure 4.1: Bounding, dilatancy and yield surface in p’-q space (a) and CSL in e-p’ space with distance between current void ratio and critical void ratio (b).

Simple anisotropic sand plasticity model

Most sophisticated implicit models are based on a critical state approach with an advanced dilatancy model and bounding surface plasticity. Especially, with faster computers and better algorithms, the use of more sophisticated models will become more important. The simple anisotropic sand plasticity model (SANISAND) was derived based on the critical state two-surface model with an open wedge yield surface in the stress space for sands by Manzari and Dafalias (1997) and the bounding surface plasticity (Dafalias and Popov, 1975; Dafalias, 1986). The bounding surface envelopes the possible stress states and the dilatancy surface separates contractive from dilative behaviour. There is an additional yield surface for the current stress state (Figure 4.1). Dilatancy and volumetric strain caused by changes in stress and void ratio are considered by means of the critical state line (CSL). The correct dilatancy behaviour is important as it relates the volumetric strain to shear strain. The plastic stiffness is considered with the distance from the centre of the yield surface to the bounding surface.

There are different versions of the SANISAND model, but all go back to Dafalias et al.

(2004). The model was successfully used for the prediction of the compaction of dry sand and the build-up of excess pore pressure for fully saturated soils in for instance earthquakes on embankment dams (Yang et al., 2020) or dynamic loading of for instance monopiles (Esfeh and Kaynia, 2020). This model is continuously developed (Taiebat and Dafalias, 2008). Liu et al. (2018b) implemented a memory surface based on the work of Corti (2016), which was also the basis for the most developed versions according to Yang et al. (2022) and Liu et al. (2021).

An overview of typical values for the SANISAND model is given in Table 4.1. The model is explained in detail in Appendix A.1.

Table 4.1: Input parameters for SANISAND04 model (Yang et al., 2020; Jostad et al., 2020; Dahl et al., 2018; Wichtmann et al., 2019; Pak et al., 2016) (cyclic values in brackets).

Symbol Ottawa F65 Toyoura Nevada NGI Montere Karlsruhe

G₀ 125 125 150 250 130 150

ν 0.05 0.05 0.05 0.05 0.05 0.05

M 1.26 1.25 1.14 1.49 1.27 1.34

c 0.735 0.712 0.78 0.6 0.712 0.7

λ 0.0287 0.019 0.027 0.013 0.02 0.112

e0 0.78 0.934 0.83 0.71 0.858 1.103

ξ 0.7 0.7 0.45 0.67 0.69 0.205

m 0.02 0.02 0.02 0.02 0.02 0.05

h0 5.0 7.05 9.7 5.0(7.0) 8.5 10.5

c_h 0.968 0.968 1.02 1.1 0.968 0.75

n_b 0.6 1.25 2.56 6.0(1.3) 1.05 1.2

A₀ 0.5 0.704 0.81 0.6 0.6 0.9

n_d 0.5 2.1 1.05 5.0(8.0) 2.5 2.0

z_max 11.0 2.0 5.0 8.0 4.0 20

cz 500 600 800 100 50.0 10000

Other implicit models

There are also many other cyclic models in addition to the SANISAND constitutive laws, whereby all models have different advantages and disadvantages, as no model can currently be considered universally applicable. One of these models is the intergranular strain anisotropy (ISA) model (Fuentes and Triantafyllidis, 2015; Wichtmann et al., 2019), which is a rate-type model and combines bounding surfaces and Karlsruhe hypoplasticity (Boulanger and Ziotopoulou, 2013; Fuentes and Triantafyllidis, 2015). The main goal of the latest ISA model is to incorporate the influence on soil response not only dependent on the void ratio and effective stress but also on the deposition method. It accounts for the inherent fabric effect by using an initial isotropic fabric structure with predominant round-shaped particles (Fuentes and Triantafyllidis, 2015). The model adopts the intergranular strain concept with which small strain effects can be captured due to the recent strain history. It uses the same dilatancy surface as the SANISAND model and requires a total of 15 parameters.

The PM4Sand model by Boulanger and Ziotopoulou (2015) simulates the response of cohesionless material under dynamic loading. It originates from Dafalias and Manzari (2004) and is an effective stress model with 21 input parameters. Most of them can be derived from practical on-site measurements such as cone penetration test (CPT) or standard penetration test results and it is mainly used in earthquake engineering.

For the consideration of liquefaction in saturated soils the University of British Columbia implemented an extension of the UBCSAND in Plaxis called UBCSAND-PLM (Puebla et al., 1997; Tsegaye, 2010; Petalas and Galavi, 2012). This constitutive model is an